All Flashcards

What is an alternating series?

A series whose terms alternate in sign.

Define convergence in the context of series.

A series converges if the sequence of its partial sums approaches a finite limit.

Define divergence in the context of series.

A series diverges if the sequence of its partial sums does not approach a finite limit.

What is $a_n$ in the context of the Alternating Series Test?

$a_n$ is the non-alternating part of the series, i.e., the terms without the $(-1)^n$ factor.

What is the general form of an alternating series?

$\sum_{n=1}^{\infty} (-1)^n a_n$ or $\sum_{n=1}^{\infty} (-1)^{n+1} a_n$ , where $a_n > 0$ for all $n$ .

State the first condition for the Alternating Series Test.

$\lim_{n \to \infty} a_n = 0$

State the second condition for the Alternating Series Test.

$a_n$ is a decreasing sequence, i.e., $a_n > a_{n+1}$ for all $n$ beyond some index.

How to test $\sum_{n=1}^{\infty} \frac{(-1)^n}{n}$ for convergence?

Identify $a_n = \frac{1}{n}$ . 2. Check $\lim_{n \to \infty} \frac{1}{n} = 0$ . 3. Verify $\frac{1}{n} > \frac{1}{n+1}$ . Since both conditions are met, the series converges.

How to test $\sum_{n=1}^{\infty} \frac{(-1)^n n}{n+1}$ for convergence?

Identify $a_n = \frac{n}{n+1}$ . 2. Check $\lim_{n \to \infty} \frac{n}{n+1} = 1 \neq 0$ . Since the limit is not 0, the series diverges.

How to test $\sum_{n=1}^{\infty} \frac{(-1)^n}{\ln(n)}$ for convergence?

Identify $a_n = \frac{1}{\ln(n)}$ . 2. Check $\lim_{n \to \infty} \frac{1}{\ln(n)} = 0$ . 3. Verify $\frac{1}{\ln(n)} > \frac{1}{\ln(n+1)}$ . Since both conditions are met, the series converges.

How to test $\sum_{n=1}^{\infty} \frac{(-1)^n n^2}{2^n}$ for convergence?

Identify $a_n = \frac{n^2}{2^n}$ . 2. Check $\lim_{n \to \infty} \frac{n^2}{2^n} = 0$ (using L'Hopital's rule). 3. Verify $\frac{n^2}{2^n} > \frac{(n+1)^2}{2^{n+1}}$ (true for large n). Since both conditions are met, the series converges.