Consider the following series. sum_{n=1}^inftyfrac{sqrt{n}+4}{n^2} The series is equivalent to the sum of two p-series. Find the value of p for each series. Determine whether the series is convergent or divergent.

iohanetc

iohanetc

Answered question

2021-03-06

Consider the following series.
n=1n+4n2
The series is equivalent to the sum of two p-series. Find the value of p for each series.
Determine whether the series is convergent or divergent.

Answer & Explanation

Arham Warner

Arham Warner

Skilled2021-03-07Added 102 answers

Given the series:
n=1n+4n2
Rewriting the given series:
n=1n+4n2=n=1nn2+4n2
n=1n+4n2=n=1nn2+n=14n2
n=1n+4n2=n=1nn212+4n=11n2
n=1n+4n2=n=1nn32+4n=11n2
Hence we get sum of two p series with:
p1=32
p2=2
Since both p1>1,p2>1, both the individual p series converge.
The given series is sum of two converging series, hence the given series n=1n+4n2 converges.
Final Answer:
p1=32
p2=2
n=1n+4n2 converges

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