5.2 Infinite Series

Introduction to Infinite Series

An infinite series is a sum of infinitely many terms. Rather than just listing numbers (that's a sequence), you're actually adding them all up. The central question is: does that sum settle on a finite value, or does it blow up?

Concept of Infinite Series Sums

An infinite series takes a sequence $\{a_n\}$ and sums its terms:

$\sum_{n=1}^{\infty} a_n = a_1 + a_2 + a_3 + \cdots$

To make sense of this, we use partial sums. The $N$th partial sum is:

$S_N = a_1 + a_2 + \cdots + a_N$

If the sequence of partial sums $\{S_N\}$ approaches a finite limit $L$ as $N \to \infty$, the series converges to $L$. If it doesn't approach any finite value, the series diverges.

This distinction matters because convergent series can stand in for exact values and be used in function approximation, integration, and solving differential equations. A divergent series, on the other hand, doesn't produce a usable sum.

Sequences vs. Series

Keep these straight, since mixing them up is a common mistake:

• A sequence is an ordered list of numbers $\{a_n\}$. You care about what happens to the individual terms as $n \to \infty$.
• A series is the sum of a sequence's terms. You care about what happens to the running total.

A sequence can converge (its terms approach a limit) while the corresponding series diverges. Classic example: $a_n = \frac{1}{n}$ converges to 0 as a sequence, but $\sum_{n=1}^{\infty} \frac{1}{n}$ (the harmonic series) diverges.

One quick test follows from this: if $\lim_{n \to \infty} a_n \neq 0$, the series $\sum a_n$ must diverge. This is the Divergence Test (also called the $n$th-Term Test). But be careful: if the limit is zero, that alone doesn't guarantee convergence. The harmonic series proves that.

Types of Infinite Series

Geometric Series

A geometric series has a constant ratio $r$ between successive terms:

$\sum_{n=0}^{\infty} ar^n = a + ar + ar^2 + ar^3 + \cdots$

where $a$ is the first term and $r$ is the common ratio.

Convergence rule:

• If $|r| < 1$, the series converges to $\dfrac{a}{1 - r}$
• If $|r| \geq 1$, the series diverges

Where the formula comes from:

1. Write the partial sum: $S_N = a + ar + ar^2 + \cdots + ar^{N-1}$

2. Multiply both sides by $r$: $rS_N = ar + ar^2 + \cdots + ar^N$

3. Subtract: $S_N - rS_N = a - ar^N$

4. Factor: $S_N(1 - r) = a(1 - r^N)$

5. Solve: $S_N = \dfrac{a(1 - r^N)}{1 - r}$

6. When $|r| < 1$, $r^N \to 0$ as $N \to \infty$, so $S_\infty = \dfrac{a}{1 - r}$

Example: Express $0.3333\ldots$ as a fraction.

Write it as $\sum_{n=1}^{\infty} 3 \cdot (0.1)^n = 0.3 + 0.03 + 0.003 + \cdots$. Here $a = 0.3$ and $r = 0.1$, so the sum is $\dfrac{0.3}{1 - 0.1} = \dfrac{0.3}{0.9} = \dfrac{1}{3}$.

Geometric series also show up in modeling exponential decay, computing present value in finance, and finding areas of self-similar geometric figures.

Telescoping Series

A telescoping series is one where most terms cancel when you write out the partial sums, leaving only a few terms at the beginning and end.

How to solve a telescoping series:

1. Decompose the general term (usually via partial fractions) into a difference of simpler expressions.
2. Write out several terms of the partial sum and observe the cancellation pattern.
3. Identify which terms survive after cancellation.
4. Take the limit of the remaining terms as $N \to \infty$.

Example: Find $\sum_{n=1}^{\infty} \dfrac{1}{n(n+1)}$

1. Partial fraction decomposition: $\dfrac{1}{n(n+1)} = \dfrac{1}{n} - \dfrac{1}{n+1}$

2. Write out the partial sum: $S_N = \left(\frac{1}{1} - \frac{1}{2}\right) + \left(\frac{1}{2} - \frac{1}{3}\right) + \left(\frac{1}{3} - \frac{1}{4}\right) + \cdots + \left(\frac{1}{N} - \frac{1}{N+1}\right)$

3. Nearly everything cancels. What's left: $S_N = 1 - \dfrac{1}{N+1}$

4. Take the limit: $\lim_{N \to \infty} S_N = 1 - 0 = 1$

The key skill with telescoping series is recognizing that a term can be split into a difference. Partial fraction decomposition is your main tool here, so stay sharp on that technique.

Power Series and Radius of Convergence

A power series is an infinite series where each term involves a power of $(x - c)$:

$\sum_{n=0}^{\infty} a_n(x - c)^n = a_0 + a_1(x - c) + a_2(x - c)^2 + \cdots$

Here $c$ is the center of the series and the $a_n$ are the coefficients.

The radius of convergence $R$ tells you how far from the center the series converges:

• For $|x - c| < R$, the series converges.
• For $|x - c| > R$, the series diverges.
• At $|x - c| = R$ (the endpoints), you have to check each case separately.

You typically find $R$ using the Ratio Test or the Root Test applied to the series terms. Power series are the foundation for Taylor and Maclaurin series, which let you represent functions like $e^x$, $\sin x$, and $\ln(1+x)$ as infinite polynomials. That representation is what makes many approximation and integration techniques possible.