# Determine if the series sum_{n=0}^infty a_n is convergent or divergent if the partial sum of the n terms of the series is given below. If the series is convergent, determine the value of the series. S_n=frac{5+8n^2}{2-7n^2}

Determine if the series $\sum _{n=0}^{\mathrm{\infty }}{a}_{n}$ is convergent or divergent if the partial sum of the n terms of the series is given below. If the series is convergent, determine the value of the series.
${S}_{n}=\frac{5+8{n}^{2}}{2-7{n}^{2}}$
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krolaniaN

Result on convergence of series:
If the sequence of partial sum ${S}_{n}={a}_{1}+{a}_{2}+\dots +{a}_{n}$ of given series $\sum _{n=0}^{\mathrm{\infty }}{a}_{n}$ is convergent sequence them the series is also convergent .
Moreover, the sum of the series is given by
$\sum _{n=0}^{\mathrm{\infty }}{a}_{n}=\underset{n\to \mathrm{\infty }}{lim}{S}_{n}$
Now, the given sequence of partial sum is
${S}_{n}=\frac{5+8{n}^{2}}{2-7{n}^{2}}$
Writing the first few terms of the sequence,
$<\frac{5+8}{2-7},\frac{5+8×{2}^{2}}{2-7×{2}^{2}},\frac{5+8×{2}^{3}}{2-7×{2}^{3}},\dots >$
$<-2.6,-1.42,-1.26,\dots >$
Clearly, the sequence is monotonically increasing.
Also it is bounded above by'0' that is ${S}_{n}<0,\mathrm{\forall }n$,
Hence, the given sequence of partial sum is convergent.
In order to get the sum of the series, we will find the limit of the given sequence of partial sum.
$\underset{n\to \mathrm{\infty }}{lim}{S}_{n}=\underset{n\to \mathrm{\infty }}{lim}\frac{5+8{n}^{2}}{2-7{n}^{2}}$
$\underset{n\to \mathrm{\infty }}{lim}{S}_{n}=\underset{n\to \mathrm{\infty }}{lim}\frac{16n}{-14n}$
$=-\frac{8}{7}$
Hence,
$\sum _{n=0}^{\mathrm{\infty }}{a}_{n}=\underset{n\to \mathrm{\infty }}{lim}{S}_{n}=\underset{n\to \mathrm{\infty }}{lim}\frac{5+8{n}^{2}}{2-7{n}^{2}}=-\frac{8}{7}$

Jeffrey Jordon