Calculus 2: Chapter 8 Essentials

Sequences, Series, and Power Series

Convergence & Divergence Tests

n-th Term Test for Divergence

If $\lim_{n\to\infty} a_n \neq 0$ or DNE, then $\sum a_n$ diverges.

Pitfall:

If $\lim_{n\to\infty} a_n = 0$, the test is inconclusive. You MUST use another test.

Geometric Series

Form: $\sum_{n=0}^{\infty} ar^n$

$$ \text{Sum} = \frac{a}{1-r} $$
  • Converges if $|r| < 1$.
  • Diverges if $|r| \geq 1$.

p-Series

Form: $\sum_{n=1}^{\infty} \frac{1}{n^p}$

  • Converges if $p > 1$.
  • Diverges if $p \leq 1$.

(Case $p=1$ is the Harmonic Series).

Ratio Test

Excellent for factorials and $n$-th powers. Compute:

$$ L = \lim_{n\to\infty} \left| \frac{a_{n+1}}{a_n} \right| $$
  • If $L < 1$, the series converges absolutely.
  • If $L > 1$ or $L=\infty$, the series diverges.
  • If $L = 1$, the test is inconclusive.

Alternating Series Test

For $\sum (-1)^n b_n$ with $b_n > 0$.

Converges if both are true:

  1. $b_{n+1} \le b_n$ (decreasing).
  2. $\lim_{n\to\infty} b_n = 0$.

Power Series

Radius & Interval of Convergence

For a series $\sum c_n (x-a)^n$ centered at $a$.

Procedure:

  1. Use the Ratio Test on the series to find an inequality involving $|x-a|$.
  2. Solve for $|x-a| < R$. This $R$ is the radius of convergence.
  3. Test the endpoints $x = a-R$ and $x = a+R$ separately by plugging them into the series and using other convergence tests (like p-series or AST).
  4. The Interval of Convergence (I.O.C) is the interval from $a-R$ to $a+R$, including any endpoints where the series converges.

Taylor & Maclaurin Series

Taylor Series (centered at $a$)

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

$ = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \dots $

Maclaurin Series (Taylor at $a=0$)

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

$ = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \dots $

Key Maclaurin Series to Memorize

Geometric Base

$$ \frac{1}{1-x} = \sum_{n=0}^{\infty} x^n $$

I.O.C: $(-1, 1)$

Exponential

$$ e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} $$

I.O.C: $(-\infty, \infty)$

Sine

$$ \sin(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!} $$

I.O.C: $(-\infty, \infty)$

Cosine

$$ \cos(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!} $$

I.O.C: $(-\infty, \infty)$

Natural Log

$$ \ln(1+x) = \sum_{n=1}^{\infty} \frac{(-1)^{n-1} x^n}{n} $$

I.O.C: $(-1, 1]$

Arctangent

$$ \arctan(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{2n+1} $$

I.O.C: $[-1, 1]$

Strategy:

Create new series from these by substituting (e.g., for $e^{2x}$, replace $x$ with $2x$), differentiating, or integrating term-by-term.