Is the sequence $\sum_{k=0}^n\frac{1}{e^k+\sqrt{k}+1} $ convergent or divergent?

Question

I have been asked to find the limit of the following sequence $$\sum_{k=0}^n\frac{1}{e^k+\sqrt{k}+1}$$

when $ n $ goes to infinity. I tried to write it as a Riemann sum but the exponential term makes it quite impossible. Any help or idea will be so appreciate. Thanks for all.

Answer 1

That the sequence converges is clear from comparison test:

$\frac{1}{e^k+\sqrt{k}+1}\leq e^{-k}$

Since the series $\sum_ke^{-k}$ converges, then so does the sequence $s_n=\sum^n_{k=1}\frac{1}{e^k+\sqrt{k}+1}$.

From the caliber of mathematics you do, I think that that may not be what you intended to ask. As for the value of the limit, I ignore whether there is closed form or an exact integral representation.

I think that a numerical computation is in place. I wrote a simple R script to estimate the first 30 elements of the sequence at double precision.

myfun <- function(x){   1/(1+ exp(x)+ sqrt(x)) }

options(digits=15) s <- sapply(0:30,function(x){myfun(x)})

data.frame(n=0:30,S_n=cumsum(s))

One of course can write simple code to obtain multiple precision, but this is just for MSE.