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<h1 class="center">Infinite Series</h1>
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<p class="center">The <b>sum</b> of infinite terms that follow a rule.</p>
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<p>When we have an infinite <a href="sequences-series.html">sequence</a> of values:</p>
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<p class="center large"><span class="intbl"><em>1</em><strong>2</strong></span> , <span class="intbl"><em>1</em><strong>4</strong></span> , <span class="intbl"><em>1</em><strong>8</strong></span> , <span class="intbl"><em>1</em><strong>16</strong></span> , ...</p>
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<p>which follow a rule (in this case each term is half the previous one),</p>
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<p class="center">and we <b>add them all up</b>:</p>
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<p class="center large"><span class="intbl"><em>1</em><strong>2</strong></span> + <span class="intbl"><em>1</em><strong>4</strong></span> + <span class="intbl"><em>1</em><strong>8</strong></span> + <span class="intbl"><em>1</em><strong>16</strong></span> + ... = S</p>
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<p class="center">we get an <b>infinite series</b>.</p>
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<div class="words">
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<p>"Series" sounds like it is the <b>list of numbers</b>, but it is actually when we add them together.</p>
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</div>
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<p class="center"><i>(Note: The dots <i>"..."</i> mean "continuing on indefinitely")</i></p>
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<h2>First Example</h2>
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<p>You might think it is impossible to work out the answer, but sometimes it can be done!</p>
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<p>Using the example from above:</p>
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<p class="center large"><span class="intbl"><em>1</em><strong>2</strong></span> + <span class="intbl"><em>1</em><strong>4</strong></span> + <span class="intbl"><em>1</em><strong>8</strong></span> + <span class="intbl"><em>1</em><strong>16</strong></span> + ... = 1</p>
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<p>And here is why:</p>
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<p class="center"><img src="images/infinite-series-1-2n.svg" alt="Sum of 1/2^n as boxes" height="208" width="208" ><br>
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(We also show a proof using Algebra below)</p>
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<h2>Notation</h2>
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<p>We often use <a href="sigma-notation.html">Sigma Notation</a> for infinite series. Our example from above looks like:</p>
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<p class="center"><img src="images/sigma-1-2n.gif" alt="Sum of 1/2^n" height="70" width="346" ></p>
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<table align="center" width="55%" border="0">
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<tbody>
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<tr>
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<td><img src="images/sigma.gif" alt="Sigma" height="34" width="32" ></td>
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<td>This symbol (called Sigma) means "sum up"</td>
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</tr>
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</tbody></table>
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<p>Try putting 1/2^n into the <a href="../numbers/sigma-calculator.html">Sigma Calculator</a>.</p>
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<h2>Another Example</h2>
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<p class="center large"><span class="intbl"><em>1</em><strong>4</strong></span> + <span class="intbl"><em>1</em><strong>16</strong></span> + <span class="intbl"><em>1</em><strong>64</strong></span> + <span class="intbl"><em>1</em><strong>256</strong></span> + ... = <span class="intbl"><em>1</em><strong>3</strong></span></p>
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<p>Each term is a quarter of the previous one, and the sum equals 1/3:</p>
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<p class="center"><img src="images/infinite-series-1-4n.svg" alt="Sum 1/4^n as boxes" height="208" width="208" ><br>
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Of the 3 spaces (1, 2 and 3) only number 2 gets filled up, hence 1/3.</p>
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<p>(By the way, this one was worked out by <i><b>Archimedes</b></i> over 2200 years ago.)</p>
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<h2>Converge</h2>
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<p>Let's add the terms one at a time. When the "sum so far" approaches a finite value, the series is said to be "<b>convergent</b>":</p>
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<div class="example">
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<p>Our first example:</p>
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<p class="center large"><span class="intbl"><em>1</em><strong>2</strong></span> + <span class="intbl"><em>1</em><strong>4</strong></span> + <span class="intbl"><em>1</em><strong>8</strong></span> + <span class="intbl"><em>1</em><strong>16</strong></span> + ...</p>
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<p>Adds up like this:</p>
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr style="text-align:center;">
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<td>Term</td>
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<td> </td>
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<td>Sum so far</td>
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</tr>
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<tr style="text-align:center;">
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<td>1/2</td>
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<td> </td>
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<td>0.5</td>
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</tr>
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<tr style="text-align:center;">
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<td>1/4</td>
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<td> </td>
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<td>0.75</td>
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</tr>
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<tr style="text-align:center;">
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<td>1/8</td>
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<td> </td>
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<td>0.875</td>
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</tr>
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<tr style="text-align:center;">
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<td>1/16</td>
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<td> </td>
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<td>0.9375</td>
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</tr>
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<tr style="text-align:center;">
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<td>1/32</td>
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<td> </td>
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<td>0.96875</td>
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</tr>
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<tr style="text-align:center;">
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<td>...</td>
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<td> </td>
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<td>...</td>
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</tr>
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</tbody></table>
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<p>The sums are heading towards a value (1 in this case), so this series is <b>convergent</b>.</p>
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</div>
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<div class="words">
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<p>The "sum so far" is called a <a href="partial-sums.html">partial sum</a> .</p>
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<p>So, more formally, we say it is a convergent series when:</p>
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<p class="center">"the sequence of partial sums has a finite <a href="../calculus/limits.html">limit</a>."</p>
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</div>
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<h2>Diverge</h2>
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<p>If the sums do not converge, the series is said to <b>diverge</b>.</p>
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<p>It can go to <b>+infinity</b>, <b>−infinity</b> or just go up and down without settling on any value.</p>
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<div class="example">
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<h3>Example:
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<div class="center large">1 + 2 + 3 + 4 + ...</div></h3>
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<p>Adds up like this:</p>
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr style="text-align:center;">
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<td>Term</td>
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<td> </td>
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<td>Sum so far</td>
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</tr>
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<tr style="text-align:center;">
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<td>1</td>
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<td> </td>
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<td>1</td>
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</tr>
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<tr style="text-align:center;">
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<td>2</td>
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<td> </td>
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<td>3</td>
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</tr>
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<tr style="text-align:center;">
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<td>3</td>
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<td> </td>
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<td>6</td>
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</tr>
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<tr style="text-align:center;">
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<td>4</td>
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<td> </td>
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<td>10</td>
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</tr>
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<tr style="text-align:center;">
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<td>5</td>
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<td> </td>
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<td>15</td>
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</tr>
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<tr style="text-align:center;">
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<td>...</td>
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<td> </td>
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<td>...</td>
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</tr>
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</tbody></table>
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<p>The sums are just getting larger and larger, not heading to any finite value.</p>
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<p>It does not converge, so it is <b>divergent</b>, and heads to infinity.</p>
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</div>
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<div class="example">
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<h3>Example: 1 − 1 + 1 − 1 + 1 ...</h3>
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<p>It goes up and down without settling towards some value, so it is <b>divergent</b>.</p>
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</div>
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<h2>More Examples</h2>
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<h3><a href="sequences-sums-arithmetic.html">Arithmetic Series</a></h3>
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<p>When the <b>difference</b> between each term and the next is a constant, it is called an <b>arithmetic series</b>.</p>
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<p class="center"><img src="images/series-arithmetic.gif" alt="Sigma n=0 to infinity of (10+2n) = 10+12+14+..." height="71" width="438" ></p>
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<p class="center">(The difference between each term is 2.)</p>
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<p> </p>
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<h3><a href="sequences-sums-geometric.html">Geometric Series</a></h3>
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<p>When the <b>ratio</b> between each term and the next is a constant, it is called a <b>geometric series</b>.</p>
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<p>Our first example from above is a geometric series:</p>
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<p class="center"><img src="images/sigma-1-2n.gif" alt="Sum of 1/2^n" height="70" width="346" ></p>
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<p class="center">(The ratio between each term is <b>½</b>)</p>
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<p>And, as promised, we can show you why that series equals 1 using Algebra:</p>
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<div class="tbl">
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<div class="row"><span class="left"><b>First</b>, we will call the whole sum <b>"S"</b>:</span><span class="right"> S = 1/2 + 1/4 + 1/8 + 1/16 + ...</span></div>
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<div class="row"><span class="left"><b>Next</b>, divide <b>S</b> by <b>2</b>:</span><span class="right">S/2 = 1/4 + 1/8 + 1/16 + 1/32 + ...</span></div>
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<p class="larger">Now <b>subtract</b> S/2 from S</p>
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<p>All the terms from 1/4 onwards cancel out.</p>
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<div class="row"><span class="left">And we get:</span><span class="right">S − S/2 = 1/2</span></div>
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<div class="row"><span class="left">Simplify:</span><span class="right"> S/2 = 1/2</span></div>
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<div class="row"><span class="left">And so:</span><span class="right">S = 1</span></div>
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</div>
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<p> </p>
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<h3>Harmonic Series</h3>
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<p>This is the Harmonic Series:</p>
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<p class="center"><img src="images/series-harmonic.gif" alt="Sigma n=1 to infinity of (1/n) = 1 + 1/2 + 1/3 + 1/4 + ..." height="70" width="397" ></p>
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<p>It is divergent.</p>
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<div class="example">
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<p>How do we know? Let's compare it to another series:</p>
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<table style="border: 0; margin:auto; text-align:center;">
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<tbody>
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<tr>
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<td>1</td>
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<td style="width:30px;">+</td>
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<td><span class="intbl"><em>1</em><strong>2</strong></span></td>
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<td style="width:30px;">+</td>
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<td><span class="intbl"><em>1</em><strong>3</strong></span>+<span class="intbl"><em>1</em><strong>4</strong></span></td>
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<td style="width:30px;">+</td>
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<td><span class="intbl"><em>1</em><strong>5</strong></span>+<span class="intbl"><em>1</em><strong>6</strong></span>+<span class="intbl"><em>1</em><strong>7</strong></span>+<span class="intbl"><em>1</em><strong>8</strong></span></td>
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<td style="width:30px;">+</td>
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<td><span class="intbl"><em>1</em><strong>9</strong></span>+...</td></tr>
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<tr>
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<td><img src="../images/style/down.svg" alt="down" height="49" width="50" ></td>
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<td><br>
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</td>
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<td><img src="../images/style/down.svg" alt="down" height="49" width="50" ></td>
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<td><br>
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</td>
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<td><img src="../images/style/down.svg" alt="down" height="49" width="50" ></td>
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<td><br>
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</td>
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<td><img src="../images/style/down.svg" alt="down" height="49" width="50" ></td>
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<td><br>
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</td>
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<td><img src="../images/style/down.svg" alt="down" height="49" width="50" ></td></tr>
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<tr>
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<td>1</td>
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<td>+</td>
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<td><span class="intbl"><em>1</em><strong>2</strong></span></td>
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<td>+</td>
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<td><span class="intbl"><em>1</em><strong>4</strong></span>+<span class="intbl"><em>1</em><strong>4</strong></span></td>
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<td>+</td>
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<td><span class="intbl"><em>1</em><strong>8</strong></span>+<span class="intbl"><em>1</em><strong>8</strong></span>+<span class="intbl"><em>1</em><strong>8</strong></span>+<span class="intbl"><em>1</em><strong>8</strong></span></td>
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<td>+</td>
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<td><span class="intbl"><em>1</em><strong>16</strong></span>+...</td></tr>
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</tbody></table>
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<p class="center">In each case, the <b>top values are equal or greater</b> than the bottom ones.</p>
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<p>Now, let's add up the bottom groups:</p>
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||
|
||
<table style="border: 0; margin:auto; text-align:center;">
|
||
<tbody>
|
||
<tr>
|
||
<td>1</td>
|
||
<td style="width:30px;">+</td>
|
||
<td><span class="intbl"><em>1</em><strong>2</strong></span></td>
|
||
<td style="width:30px;">+</td>
|
||
<td><span class="intbl"><em>1</em><strong>4</strong></span>+<span class="intbl"><em>1</em><strong>4</strong></span></td>
|
||
<td style="width:30px;">+</td>
|
||
<td><span class="intbl"><em>1</em><strong>8</strong></span>+<span class="intbl"><em>1</em><strong>8</strong></span>+<span class="intbl"><em>1</em><strong>8</strong></span>+<span class="intbl"><em>1</em><strong>8</strong></span></td>
|
||
<td style="width:30px;">+</td>
|
||
<td><span class="intbl"><em>1</em><strong>16</strong></span>+...</td>
|
||
<td style="width:30px;"><br>
|
||
</td>
|
||
<td><br>
|
||
</td></tr>
|
||
<tr>
|
||
<td><img src="../images/style/down.svg" alt="down" height="49" width="50" ></td>
|
||
<td><br>
|
||
</td>
|
||
<td><img src="../images/style/down.svg" alt="down" height="49" width="50" ></td>
|
||
<td><br>
|
||
</td>
|
||
<td><img src="../images/style/down.svg" alt="down" height="49" width="50" ></td>
|
||
<td><br>
|
||
</td>
|
||
<td><img src="../images/style/down.svg" alt="down" height="49" width="50" ></td>
|
||
<td><br>
|
||
</td>
|
||
<td><img src="../images/style/down.svg" alt="down" height="49" width="50" ></td>
|
||
<td><br>
|
||
</td>
|
||
<td><br>
|
||
</td></tr>
|
||
<tr>
|
||
<td>1</td>
|
||
<td>+</td>
|
||
<td><span class="intbl"><em>1</em><strong>2</strong></span></td>
|
||
<td>+</td>
|
||
<td><span class="intbl"><em>1</em><strong>2</strong></span></td>
|
||
<td>+</td>
|
||
<td><span class="intbl"><em>1</em><strong>2</strong></span><span class="intbl"><strong></strong></span></td>
|
||
<td>+</td>
|
||
<td><span class="intbl"><em>1</em><strong>2</strong></span></td>
|
||
<td>+</td>
|
||
<td>... = ∞</td></tr>
|
||
</tbody></table>
|
||
<p class="center"><b>That</b> series is divergent.</p>
|
||
<p>So the <b>harmonic series</b> must also be divergent.</p>
|
||
</div>
|
||
<p>Here is another way:</p>
|
||
|
||
<div class="example">
|
||
<p>We can sketch the area of each term and compare it to the area under the <b>1/x</b> curve:</p>
|
||
<p class="center"><img src="images/series-harmonic-graph.svg" alt="harmonic series graph" height="181" width="263" ><br>
|
||
<span class="larger"><b>1/x</b> vs harmonic series area</span></p>
|
||
<p>Calculus tells us the area under 1/x (from 1 onwards) approaches <b>infinity</b>, and the harmonic series is greater than that, so it must be divergent.</p>
|
||
</div>
|
||
<p> </p>
|
||
|
||
<h3>Alternating Series</h3>
|
||
<p>An Alternating Series has terms that alternate between positive and negative.</p>
|
||
<p>It may or may not converge.</p>
|
||
|
||
<div class="example">
|
||
|
||
<h3>Example: <span class="intbl"><em>1</em><strong>2</strong></span> − <span class="intbl"><em>1</em><strong>4</strong></span> + <span class="intbl"><em>1</em><strong>8</strong></span> − <span class="intbl"><em>1</em><strong>16</strong></span> + ... = <span class="intbl"><em>1</em><strong>3</strong></span></h3>
|
||
<p>This illustration may convince you that the terms converge on <span class="intbl"><em>1</em><strong>3</strong></span>:</p>
|
||
<p class="center"><img src="images/infinite-series-1-2m1-4.svg" alt="harmonic series graph" height="212" width="364" ></p>
|
||
<p>Maybe you can try to prove it yourself? Try pairing up each plus and minus pair, then look up above for a series that matches.</p>
|
||
</div>
|
||
<p>Another example of an Alternating Series (based on the Harmonic Series above):</p>
|
||
<p class="center"><img src="images/series-alternating.gif" alt="Sigma n=1 to infinity of (-1)^(n+1) /n = 1 - 1/2 + 1/3 - 1/4 + ... = ln(2)" height="71" width="473" ></p>
|
||
<p>This one converges on the natural <a href="logarithms.html">logarithm</a> of 2</p>
|
||
|
||
<div class="example">
|
||
|
||
<h3>Advanced Explanation:</h3>
|
||
<p>To show WHY, first we start with a square of area 1, and then pair up the minus and plus fractions to show how they cut the area down to the area under the curve <b>y=1/x</b> between 1 and 2:</p>
|
||
<p class="center"><img src="images/alternating-harmonic.svg" alt="alternating harmonic proof" height="160" width="617" ></p>
|
||
<p>Can you see what remains is the area of 1/x from 1 to 2?</p>
|
||
<p>Using <a href="../calculus/integration-introduction.html">integral calculus</a> (trust me) that area is <b>ln(2)</b>:</p>
|
||
<div class="center">
|
||
<div class="intgl">
|
||
<div class="to">2</div>
|
||
<div class="symb"></div>
|
||
<div class="from">1</div>
|
||
</div>1/x dx = ln(2) − ln(1) = <b>ln(2)</b></div>
|
||
</div>
|
||
<p> </p>
|
||
<p><b>You</b> can investigate this further!</p>
|
||
<ul>
|
||
<li>do those rectangles really make the area above the curve as shown?</li>
|
||
<li>is the area below the curve really ln(2) = 0.693... ? Try the <a href="../calculus/integral-approximation-calculator.html">Integral Approximation Calculator</a> to see (y=1/x between 1 and 2)</li>
|
||
</ul>
|
||
|
||
|
||
<h2>Order!</h2>
|
||
|
||
<p>The order of the terms can be very important! We can sometimes get weird results when we change their order.</p>
|
||
<p>For example in an alternating series, what if we made all positive terms come first? So be careful!</p>
|
||
|
||
|
||
<h2>More</h2>
|
||
|
||
<p>There are other types of Infinite Series, and it is interesting (and often challenging!) to work out if they are convergent or not, and what they may converge to.</p>
|
||
<p> </p>
|
||
|
||
<div class="related">
|
||
<a href="sigma-notation.html">Sigma Notation</a>
|
||
<a href="../numbers/sigma-calculator.html">Sigma Calculator</a>
|
||
<a href="partial-sums.html">Partial Sums</a>
|
||
<a href="index.html">Algebra Index</a>
|
||
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|
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