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. This is often easier than the Direct Comparison Test.

Alternating Series & Absolute Convergence

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The Alternating Series Test

An alternating series $\sum (-1)^n b_n$ (with $b_n > 0$) converges if both conditions are met:

  1. $b_{n+1} \le b_n$ for all large $n$ (the terms are eventually decreasing).
  2. $\lim_{n \to \infty} b_n = 0$.

Absolute vs. Conditional Convergence

  • A series $\sum a_n$ is absolutely convergent if the series of absolute values, $\sum |a_n|$, converges. (This is a stronger form of convergence).
  • A series $\sum a_n$ is conditionally convergent if it converges, but $\sum |a_n|$ diverges. The Alternating Harmonic Series $\sum \frac{(-1)^{n-1}}{n}$ is the classic example.

Key Fact: If a series converges absolutely, then it converges.

The Ratio & Root Tests

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These tests are powerful for series involving factorials ($n!$) or $n$-th powers.

Ratio Test

Let $L = \lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right|$.

  • If $L < 1$, the series is absolutely convergent.
  • If $L > 1$ or $L = \infty$, the series is divergent.
  • If $L = 1$, the test is inconclusive.

Root Test

Let $L = \lim_{n \to \infty} \sqrt[n]{|a_n|}$.

  • The conclusions are identical to the Ratio Test based on the value of $L$.

Power, Taylor, & Maclaurin Series

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A Power Series is a series of the form $\sum c_n(x-a)^n$. Our goal is to find its Radius of Convergence (R) and Interval of Convergence (I.O.C.).

A Taylor Series is a way to represent a function $f(x)$ as a power series centered at $x=a$. A Maclaurin Series is a Taylor Series centered at $a=0$.

Taylor Series Formula

$$ f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x-a)^n $$

Key Maclaurin Series to Memorize

  • $e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \dots$
  • $\sin(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \dots$
  • $\cos(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!} = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \dots$
  • $\frac{1}{1-x} = \sum_{n=0}^{\infty} x^n = 1 + x + x^2 + \dots$

Partial Sums Visualizer

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See how a series behaves by watching its sequence of partial sums ($S_k$) evolve. Select a series and watch the points $(k, S_k)$ get plotted.