Infinite Sequences and Series

Taylor Series (centered at $c$)

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

Maclaurin Series

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

Maclaurin Series for $e^x$

$$e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots$$

Maclaurin Series for $\sin x$

$$\sin x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots$$

Maclaurin Series for $\cos x$

$$\cos x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!} = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots$$

Maclaurin Series for $\frac{1}{1-x}$

$\frac{1}{1-x} = \sum_{n=0}^{\infty} x^n = 1 + x + x^2 + \cdots$, |x| < 1

Sum of infinite geometric series

$S = \frac{a}{1-r}$, valid when |r| < 1

nth Term Test (Divergence Test)

If $\lim_{n\to\infty} a_n \neq 0$, the series diverges. If $= 0$, inconclusive.

Geometric Series Test

$\sum ar^n$ converges to $\frac{a}{1-r}$ when |r|<1; diverges when $|r|\geq 1$

p-Series Test

$\sum \frac{1}{n^p}$ converges if p > 1; diverges if $p \leq 1$

Integral Test

If $f$ is positive, continuous, and decreasing, then $\sum a_n$ and $\int f(x)\,dx$ both converge or both diverge

Direct Comparison Test

If $0 \leq a_n \leq b_n$: $\sum b_n$ converges $\Rightarrow$ $\sum a_n$ converges; $\sum a_n$ diverges $\Rightarrow$ $\sum b_n$ diverges

Limit Comparison Test

If \lim \frac{a_n}{b_n} = L > 0 (finite), then $\sum a_n$ and $\sum b_n$ both converge or both diverge

Alternating Series Test

$\sum(-1)^n b_n$ converges if (1) $b_n$ is decreasing and (2) $\lim_{n\to\infty} b_n = 0$

Alternating Series Estimation Theorem

$$|\text{Error}| \leq b_{n+1}$$

Ratio Test

$\lim\left|\frac{a_{n+1}}{a_n}\right| < 1 \Rightarrow$ converges; $> 1 \Rightarrow$ diverges; $= 1 \Rightarrow$ inconclusive

Radius of Convergence

$R = \frac{1}{\lim|a_{n+1}/a_n|}$; series converges for |x-c| < R

Absolute vs. Conditional Convergence

Absolutely convergent: $\sum|a_n|$ converges. Conditionally convergent: $\sum a_n$ converges but $\sum|a_n|$ diverges.

Lagrange Error Bound

$|f(x)-P_n(x)| \leq \frac{M|x-c|^{n+1}}{(n+1)!}$, where $M \geq |f^{(n+1)}(t)|$ for all $t$ between $x$ and $c$