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Calc II: Series & Power Series [Stealth Mode]

Calc II: Series & Power Series [Stealth Mode]











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 $p \le 1$.…