Lesson 14 - Infinite Series

Analyzing series is important because it allows us to evaluate, differentiate, and integrate complicated functions by using polynomials that are easier to handle.

Sums and Series

An infinite series is a sum of infinitely many terms and is written in the form

\sum_{n = 1}^{\infty} a_{n} = a_{1} + a_{2} + a_{3} \dots

However, we can't add an infinite number of terms in the same way we can add a finite number of terms. Instead, the value of an infinite series is defined in terms of the limit of partial sums. A partial sum of an infinite series is a finite sum of the form

\sum_{n = 1}^{k} a_{n} = a_{1} + a_{2} + a_{3} + \dots

Example

Suppose oil is seeping into a lake such that 1000 gallons enters the lake the first week. During the second week, an additional 500 gallons of oil enters the lake. The third week, 250 more gallons enters the lake. Assume this pattern continues such that each week half as much oil enters the lake as did the previous week. If this continues forever, what can we say about the amount of oil in the lake? Will the amount of oil continue to get arbitrarily large, or is it possible that it approaches some finite amount? To answer this question, we look at the amount of oil int he lake after $k$ weeks. Letting $S_{k}$ denote the amount of oil in the lake (measured in thousands of gallons after $k$ weeks), we see that

\begin{array}{l} S_{1} = 1 \\ S_{2} = 1 + 0.5 = 1 + \frac{1}{2} \\ S_{3} = 1 + 0.5 + 0.125 = 1 + \frac{1}{2} + \frac{1}{4} \\ S_{4} = 1 + 0.5 + 0.125 + 0.125 = 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} \\ S_{5} = 1 + 0.5 + 0.25 + 0.125 + 0.0625 = 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} \end{array}

Looking at this patter, we see that the amount of oil in the lake after k weeks is

S_{k} = 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \dots + \frac{1}{2^{k - 1}} = \sum_{n = 1}^{k} {(\frac{1}{2})}^{n - 1}

We are interested in what happens as $k \to \infty$ . Symbolically, the amount of oil in the lake as $k \to \infty$ is given by the infinite series

\sum_{n = 1}^{\infty} = 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{6} + \dots

At the same time, as $k \to \infty$ , the amount of oil in the lake can be calculated by evaluating $lim_{k \to \infty} S_{k}$ . We want to see if $S_{k}$ diverges or converges. This will tell us if the series converges.

If we sum $S$ for each value $k$ , we get

\begin{array}{l} S_{1} = 1 \\ S_{2} = 1 + \frac{1}{2} = \frac{3}{2} \\ S_{3} = 1 + \frac{1}{2} + \frac{1}{4} = \frac{7}{4} \\ S_{4} = 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} = \frac{15}{8} \\ S_{5} = 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} = \frac{31}{16} \end{array}

If we plot these values, we get something like this.

Screenshot 2024-10-25 at 12.45.57 PM.png

The values look to converge on the value 2.

We can say that

\sum_{n = 1}^{\infty} {(\frac{1}{2})}^{n - 1} = 2

Let's generalize a formal definition.

Definition

An infinite series is an expression of the form

\sum_{n = 1}^{\infty} a_{n} = a_{1} + a_{2} + a_{3} + \dots

For each positive integer $k$ , the sum

S_{k} = \sum_{n = 1}^{k} a_{n} = a_{1} + a_{2} + a_{3} + \dots + a_{k}

is called the $k$ th partial sum of the infinite series. The partial sums form a sequence $S_{k}$ . If the sequence of partial sums converges to a real number $S$ , the infinite series converges. If we can describe the convergence of a series to $S$ , we call $S$ the sum of the series, and we write

\sum_{n = 1}^{\infty} a_{n} = S

If the sequence of partial sums diverges, we have the divergence of a series.

For this class, we're going to look at a specific series. Let's get into that in the next lesson.

Next Lesson: Lesson 15 - Geometric Series and Ratio Test

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Note: This lesson adapted from OpenStax. Credit goes to OpenStax creators.