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Calculus II - Infinite Series Deep Dive

Calculus II - Infinite Series Deep Dive















Calculus II

Infinite Series Deep Dive






Series Fundamentals

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An infinite series is the sum of the terms of an infinite sequence $\{a_n\}$. The core question is whether this sum approaches a finite value (converges) or not (diverges).


Sequence of Partial Sums

The convergence of a series $\sum a_n$ is defined by the convergence of its sequence of partial sums, $S_k = \sum_{n=1}^{k} a_n = a_1 + a_2 + \dots + a_k$.

If $\lim_{k \to \infty} S_k = L$ (a finite number), the series converges to $L$. Otherwise, it diverges.




Geometric Series: The Foundation

A series of the form $\sum_{n=1}^{\infty} ar^{n-1}$. This is one of the few series where we can easily find the sum.


Converges if $|r| < 1$. The sum is $S = \frac{a}{1-r}$, where 'a' is the first term.

Diverges if $|r| \geq 1$.





The Test for Divergence

This should always be your first thought! For a series $\sum a_n$:

If $\lim_{n \to \infty} a_n \neq 0$ or the limit DNE, the series **diverges**.

Warning: If $\lim_{n \to \infty} a_n = 0$, this test is inconclusive. The series might converge or diverge. (e.g., The Harmonic Series $\sum \frac{1}{n}$ diverges, but its terms go to 0).











Tests for Positive Series

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These tests apply to series with non-negative terms.


The Integral Test: If $f(x)$ is positive, continuous, and decreasing for $x \ge 1$ and $a_n = f(n)$, then $\sum a_n$ and $\int_1^\infty f(x)dx$ either both converge or both diverge.

The p-Series Test: A direct result of the Integral Test. The series $\sum \frac{1}{n^p}$ converges if $p > 1$ and diverges if $p \leq 1$.

Direct Comparison Test: If $0 \le a_n \le b_n$, if the "bigger" series $\sum b_n$ converges, so does $\sum a_n$. If the "smaller" series $\sum a_n$ diverges, so does $\sum b_n$.

Limit Comparison Test: For two positive series $\sum a_n$ and $\sum b_n$, if $\lim_{n \to \infty} \frac{a_n}{b_n} = L$ where $L$ is finite and positive ($0 < L < \infty$), then both series either converge or diverge together.…