Finding the summation of this peculiar type of infinite series

Asked Feb 05 '23 at 15:22

Active Feb 05 '23 at 15:22

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Find $\Sigma_{r=1}^{\infty} \frac{1}{a+\frac{b^{r}}{r}}$ where $b>1$ and $a$ is a positive real number.

I guess that this sum must be convergent as the terms gets smaller, but I have no idea on where to start on.

I tried using differentiation of $\frac{1}{1-x}$ as a geometric progression, but that lead me nowhere. I have also learnt that definite integrals can be represented as limits of infinite sums.

Can I use a definite integral to evaluate this summation as a Riemann summation? Is there a closed form for this?

Any help would be appreciated. Thanks in advance.

asked Feb 05 '23 at 15:22

Rishi Shekher

1

Doesn't the series diverge since $\frac1{a+b^r/r}\to\frac1a>0$ as $r\to\infty$? The successive terms need to approach $0$ for the sum to converge; merely decreasing isn't sufficient. – Phobo Havuz Feb 05 '23 at 15:37
It converges for some values and the $b=2,a=1$ case has no closed form found. You could get an integral representation – Тyma Gaidash Feb 05 '23 at 15:46
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@Phobo Havuz How does this diverge? If b> 1, b^r/r tends to infinity and the terms tend to 0. – Rishi Shekher Feb 06 '23 at 03:35
Oops yeah you're right, it does converge for some $b>1$. Please ignore my previous comment – Phobo Havuz Feb 06 '23 at 09:20
@TymaGaidash. Could you give an idea about a possible integral representation ? Thanks & cheers :-) – Claude Leibovici Feb 07 '23 at 09:22
@ClaudeLeibovici Not much has helped, but the sum is almost like one here. Do you have any ideas? – Тyma Gaidash Feb 07 '23 at 13:01

Finding the summation of this peculiar type of infinite series

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