I am unable to prove a deduction which is to be used in proving Theorem 11.16 in Apostol introduction to analytic number theory.
I am adding image of result which I am unable to prove. Can someone please give a hint!!
2026-04-03 15:12:05.1775229125
Doubt in a deduction to be used in proving a result in Analytic number theory
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HINT
$\sum_0^\infty x^{n+1}=\frac{x}{1-x}$.
Differentiate, multiply by $x$ then differentiate again.
However, the middle term in Apostol's answer is wrong, it should be minus his answer.