Convergence of series \(\displaystyle{\sum_{{{n}={1}}}^{\infty}}{\ln{{\left({\frac{{{2}{n}+{7}}}{{{2}{n}+{1}}}}\right)}}}\) I have the

Reuben Brennan

Reuben Brennan

Answered question

2022-03-25

Convergence of series n=1ln(2n+72n+1)
I have the series
n=1ln(2n+72n+1)
I'm trying to find if the sequence converges and if so, find its sum.
I have done the ratio and root test but It seems it is inconclusive.
How can I find if the series converges?

Answer & Explanation

Ireland Vaughan

Ireland Vaughan

Beginner2022-03-26Added 14 answers

No need for convergence tests! Note that if f(n)=ln(2n+72n+1) then:
f(n)+f(n+3)=ln(2n+13)ln(2n+1)
So most terms cancel out. In other words, the partial sum of your series is:
1mf(n)=ln3ln5ln7+ln(2m+3)+ln(2m+5)+ln(2m+7)
=ln(2m+3)(2m+5)(2m+7)ln105
Which obviously does not converge.

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