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}

mattgondek4 2020-11-23 Answered
Determine if the series n=0an 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.
Sn=5+8n227n2
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Expert Answer

krolaniaN
Answered 2020-11-24 Author has 86 answers

Result on convergence of series:
If the sequence of partial sum Sn=a1+a2++an of given series n=0an is convergent sequence them the series is also convergent .
Moreover, the sum of the series is given by
n=0an=limnSn
Now, the given sequence of partial sum is
Sn=5+8n227n2
Writing the first few terms of the sequence,
<5+827,5+8×2227×22,5+8×2327×23,>
<2.6,1.42,1.26,>
Clearly, the sequence is monotonically increasing.
Also it is bounded above by'0' that is Sn<0,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.
limnSn=limn5+8n227n2
limnSn=limn16n14n
=87
Hence,
n=0an=limnSn=limn5+8n227n2=87

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Jeffrey Jordon
Answered 2022-01-14 Author has 2262 answers

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