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Explain the nth Term Test for Divergence.

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

Explain the Limit Comparison Test.

If $\lim_{n \to \infty} \frac{a_n}{b_n} = c$ , where $c$ is finite and positive, then $\sum a_n$ and $\sum b_n$ either both converge or both diverge.

Explain the Direct Comparison Test.

If $0 \leq a_n \leq b_n$ and $\sum b_n$ converges, then $\sum a_n$ converges. If $a_n \geq b_n \geq 0$ and $\sum b_n$ diverges, then $\sum a_n$ diverges.

Explain the Integral Test.

If $f(x)$ is continuous, positive, and decreasing for $x \geq 1$ , then $\sum_{n=1}^{\infty} f(n)$ and $\int_{1}^{\infty} f(x) dx$ either both converge or both diverge.

Explain the Alternating Series Test.

If $a_n$ is decreasing and $\lim_{n \to \infty} a_n = 0$ , then the alternating series $\sum (-1)^n a_n$ converges.

Explain the Ratio Test.

If $\lim_{n \to \infty} |\frac{a_{n+1}}{a_n}| = L$ , then the series converges if $L < 1$ , diverges if $L > 1$ , and is inconclusive if $L = 1$ .

Explain the concept of radius of convergence.

The distance from the center of a power series within which the series converges.

Explain the Alternating Series Error Bound.

The error in approximating an alternating series is less than or equal to the absolute value of the first omitted term.

Explain the Lagrange Error Bound.

The error in approximating a function using a Taylor polynomial is bounded by the Lagrange Error Bound, which depends on the (n+1)th derivative.

What does it mean for a series to converge absolutely?

A series $\sum a_n$ converges absolutely if $\sum |a_n|$ converges.

What does it mean for a series to converge conditionally?

A series $\sum a_n$ converges conditionally if $\sum a_n$ converges but $\sum |a_n|$ diverges.

What is the formula for the nth term of an arithmetic sequence?

$a + (n-1)d$ , where $a$ is the first term and $d$ is the common difference.

What is the formula for the nth term of a geometric sequence?

$a * r^{n-1}$ , where $a$ is the first term and $r$ is the common ratio.

What is the general form of a power series?

$\sum a_n(x-c)^n$

What is the Maclaurin series for $e^x$ ?

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

What is the Maclaurin series for $\sin(x)$ ?

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

What is the Maclaurin series for $\cos(x)$ ?

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

What is the formula for Lagrange Error Bound?

$|R_n(x)| \leq \frac{M}{(n+1)!}|x-c|^{n+1}$ , where M is the maximum value of the (n+1)th derivative.

What is the formula for Alternating Series Error Bound?

$|Error| \leq |a_{n+1}|$ , where $a_{n+1}$ is the (n+1)th term of the series.

What does the Alternating Series Test state?

If $a_n$ is decreasing and $\lim_{n \to \infty} a_n = 0$ , then the alternating series $\sum (-1)^n a_n$ converges.

What does the Ratio Test state?

If $\lim_{n \to \infty} |\frac{a_{n+1}}{a_n}| = L$ , then the series converges if $L < 1$ , diverges if $L > 1$ , and is inconclusive if $L = 1$ .

What does the nth Term Test for Divergence state?

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