# Determinate the convergence or divergence of sum_(n=0)^oo((-1)^n)/(sqrt(n+1))

Determinate the convergence or divergence of $\sum _{n=0}^{\mathrm{\infty }}\frac{\left(-1{\right)}^{n}}{\sqrt{n+1}}$
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autarhie6i
$\sum _{n=0}^{\mathrm{\infty }}\frac{\left(-1{\right)}^{n}}{\sqrt{n+1}}\phantom{\rule{0ex}{0ex}}{a}_{n}=\frac{1}{\sqrt{n+1}}>0\phantom{\rule{0ex}{0ex}}\underset{n\to \mathrm{\infty }}{lim}{a}_{n}=\underset{n\to \mathrm{\infty }}{lim}\frac{1}{\sqrt{n+1}}\phantom{\rule{0ex}{0ex}}\frac{1}{\mathrm{\infty }}\phantom{\rule{0ex}{0ex}}\underset{n\to \mathrm{\infty }}{lim}=0$
$1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\frac{1}{\sqrt{4}}+\cdots$
is clearly ${a}_{n+1}\le {a}_{n}{\mathrm{\forall }}_{n}>N$
Hence By alternating Series Test
$\sum _{n=0}^{\mathrm{\infty }}\frac{\left(-1{\right)}^{n}}{\sqrt{n+1}}$ is convergent