## Related questions with answers

State the following. The Alternating Series Test

Solutions

Verified$\textbf{Alternating Series Test}$

$\text{The alternating series $\sum_{n=k}^{\infty} a_n$ is convergent if the following two conditions are met: }$

(1) $\lim\limits_{n \to \infty}|a_n|=0$

(2) $|a_{n+1}|\leq|a_n|$

The Alternating Series Test states that: If $\sum a_n$ is an alternating series and satisfies the following two conditions: (i) $|a_{n+1}| \leq |a_{n}|$ for all $n$, and (ii) $\lim_{n \rightarrow \infty}{|a_n|} = 0$, then the series converges. In words, condition (i) means the absolute value of the terms should be getting smaller, and how small they should be getting is indicated by condition (ii): they should be getting to 0. If one or both of these conditions fail, then the series diverges.

$\textbf{The Alternating Series Test:}$

Let $\sum (-1)^n \, b_n$ be an alternating series such that $b_n > 0$ for all $n$. If the following is satisfied:

$\begin{align*} (i) \quad &\quad \quad b_{n+1} \leq b_n \\ (ii) \quad &\quad \quad \lim_{n\to\infty} b_n = 0 \end{align*}$

then the alternating series is convergent.

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