I am given the series \sum_{n=1}^\infty\frac{n^{n-1}}{(2n^2+n+1)^{(n+\frac{1}{2})}}

Lubna Boyce

Lubna Boyce

Answered question

2022-02-28

I am given the series
n=1nn1(2n2+n+1)(n+12)

Answer & Explanation

shotokan0758s

shotokan0758s

Beginner2022-03-01Added 8 answers

The ratio test works fine
an=nn1(2n2+n+1)(n+12)
Take loragithms
log(an)=(n1)log(n)(n+12)log(2n2+n+1)
Using Taylor for large values of n.
log(an)=nlog(12n)+12(log(12n4)1)58n+O(1n2)
Apply it twice and continue with Taylor series
log(an+1)log(an)=(log(12n)1)52n
an+1an=elog(an+1)log(an)=12en(152en+O(1n2))
Now, using the expansion of log(an) to O(1n), we then have your formula
an12e12nnn+2
which is an overestimate of the true an

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?