Calculus II Study Guide

Sequences, Series, and Power Series

1. Sequences: The Foundation

A sequence is just an infinite list of numbers, $\{a_n\}$. The main question we ask is: Where are these numbers heading?

A sequence $\{a_n\}$ converges to a limit $L$ if the terms get closer and closer to $L$ as $n$ gets huge.

If $\lim_{n \to \infty} a_n = L$ (and $L$ is a finite number), the sequence converges.
If the limit is $\infty$, $-\infty$, or does not exist, the sequence diverges.

2. Series: The Sum of Terms

A series is what you get when you try to add up all the terms of a sequence: $\sum a_n$. The big question is: Does this infinite sum add up to a finite number?

The Test for Divergence (The First Check!)

This is your first line of defense. It's a quick check to see if a series is obviously divergent.

Take the limit of the terms: $\lim_{n \to \infty} a_n$.
If $\lim_{n \to \infty} a_n \neq 0$, the series $\sum a_n$ DIVERGES.
Important Note: If the limit IS 0, this test tells you NOTHING. The series might converge or it might diverge. You must use another test.

3. The Arsenal: Tests for Convergence

This is your toolbox. Your job is to pick the right tool for the series you're given.

Geometric Series Test

When to use: When you see a number raised to the power of $n$, like $\sum ar^n$.

For a series of the form $\sum ar^n$:
  • Converges if $|r| < 1$. The sum is $S = \frac{a}{1-r}$ ($a$ = first term).
  • Diverges if $|r| \ge 1$.

Integral Test

When to use: When $a_n$ can be easily turned into a function $f(x)$ that you can integrate (and is positive, continuous, and decreasing).

The series $\sum a_n$ and the improper integral $\int_{1}^{\infty} f(x) dx$ either both converge or both diverge.

p-Series Test

When to use: A shortcut for series of the form $\sum \frac{1}{n^p}$. This is your most important benchmark for comparison tests!

The series $\sum_{n=1}^{\infty} \frac{1}{n^p}$:
  • Converges if $p > 1$.
  • Diverges if $p \le 1$. (The case $p=1$ is the Harmonic Series, which always diverges).

Limit Comparison Test

When to use: For messy-looking rational functions or anything that "looks like" a p-series.

Pick a simpler series $\sum b_n$ that looks like $\sum a_n$. Compute the limit: $$L = \lim_{n \to \infty} \frac{a_n}{b_n}$$

If $L$ is a finite, positive number, then both series do the same thing (both converge or both diverge).

Alternating Series Test

When to use: When you see alternating signs like $(-1)^n$ or $(-1)^{n+1}$.

An alternating series $\sum (-1)^n b_n$ converges if BOTH conditions are met:
  1. The terms are decreasing: $b_{n+1} \le b_n$.
  2. The limit of the terms is zero: $\lim_{n \to \infty} b_n = 0$.

Ratio Test

When to use: The best choice for series with factorials ($n!$) and exponentials ($c^n$).

Compute the limit: $$L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|$$
  • If $L < 1$, the series converges absolutely.
  • If $L > 1$, the series diverges.
  • If $L = 1$, the test is INCONCLUSIVE. Try another test!

4. Power Series: Functions as Infinite Polynomials

A power series is a series with a variable, $x$. The goal is to find which $x$ values make the series converge.

A power series centered at $a$: $$\sum_{n=0}^{\infty} c_n (x-a)^n$$

Finding the Radius & Interval of Convergence

This is a 3-step process:

  1. Use the Ratio Test. You'll get an inequality like $|x-a| < R$. This $R$ is the Radius of Convergence.
  2. The series converges for $x$ in the open interval $(a-R, a+R)$.
  3. Test the endpoints! Plug $x = a-R$ and $x = a+R$ into the series and use another test to check for convergence. This gives the final Interval of Convergence.

5. Representing Functions as Power Series

We can express common functions as power series. This is an incredibly powerful tool.

Key Maclaurin Series to Memorize

1. $\frac{1}{1-x} = \sum_{n=0}^{\infty} x^n = 1 + x + x^2 + \dots$ for $|x| < 1$
2. $e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \dots$ for all $x$
3. $\sin(x) = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n+1}}{(2n+1)!} = x - \frac{x^3}{3!} + \dots$ for all $x$
4. $\cos(x) = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n}}{(2n)!} = 1 - \frac{x^2}{2!} + \dots$ for all $x$
5. $\ln(1+x) = \sum_{n=1}^{\infty} (-1)^{n-1} \frac{x^n}{n} = x - \frac{x^2}{2} + \dots$ for $-1 < x \le 1$

6. Taylor and Maclaurin Series: The General Formula

If you can't build a series from the list above, you can use the definition. A Maclaurin series is a Taylor series centered at $a=0$.

The Taylor series for $f(x)$ centered at $a$ is: $$f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x-a)^n$$

> Compute derivatives of $f(x)$, plug in $a$, find the pattern, and build the series.

7. Final Advice & Common Pitfalls