Answer
Convergent
Work Step by Step
Alternating series test:
Suppose that we have series $\Sigma a_n$, such that $a_{n}=(-1)^{n}b_n$ or $a_{n}=(-1)^{n+1}b_n$, where $b_n\geq 0$ for all $n$.
Then if the following two condition are satisfied the series is convergent.
1. $\lim\limits_{n \to \infty}b_{n}=0$
2. $b_{n}$ is a decreasing sequence.
Given: $\Sigma_{n=0}^{\infty}\frac{(-1)^{n+1}}{\sqrt {n+1}}$
In the given problem, $b_{n}=\frac{1}{\sqrt {n+1}}$
which satisfies both conditions of Alternating Series Test as follows:
1. $b_{n}=\frac{1}{\sqrt {n+1}}$is decreasing because the denominator is increasing.
2. $\lim\limits_{n \to \infty}b_{n}=\lim\limits_{n \to \infty}\frac{1}{\sqrt {n+1}}=\frac{1}{\infty}=0$
Hence, the given series is convergent by Alternating Series Test.
