Trying to evaluate $\lim\limits_{n\in\mathbb{N}\to\infty}\left[n+\sum\limits_{k=1}^{\infty}\frac{(-1)^k}{k!}\frac{b^{nk}}{b^k-1}\right]$

Question

I need some help to evaluate this limit : $$\lim\limits_{n\in\mathbb{N}\to\infty}\left[n+\sum\limits_{k=1}^{\infty}\frac{(-1)^k}{k!}\frac{b^{nk}}{b^k-1}\right]$$ With $b\in\left(1,+\infty\right)$.

If it's not possible to get a closed form for unspecified $b$, then I'd still gladly take any value for any particular case of $b$...

Now this looks like a hard one to me, so I'm curious what people here can get out of it !

Answer 1

Write

$$ \sum_{k=1}^{\infty}\frac{(-1)^k}{k!}\frac{b^{nk}}{b^k-1} = \sum_{k=1}^{\infty}\frac{(-1)^k}{k!} \sum_{j=1}^{\infty} b^{(n-j)k} = \sum_{j=1}^{\infty} \left( e^{-b^{n-j}} - 1 \right). $$

Then

\begin{align*} n + \sum_{k=1}^{\infty}\frac{(-1)^k}{k!}\frac{b^{nk}}{b^k-1} &= n + \sum_{j=1}^{n} \left( e^{-b^{n-j}} - 1 \right) + \sum_{j=n+1}^{\infty} \left( e^{-b^{n-j}} - 1 \right) \\ &= \sum_{p=0}^{n-1} e^{-b^{p}} + \sum_{p=1}^{\infty} \left( e^{-b^{-p}} - 1 \right), \end{align*}

Taking $n\to\infty$, this converges to

$$ \sum_{p=0}^{\infty} e^{-b^{p}} + \sum_{p=1}^{\infty} \left( e^{-b^{-p}} - 1 \right) $$

I will be surprised if this has a closed form in elementary functions.

Answer 2

Looking at Sangchul Lee's answer, I very vaguely remember (problem of age !) that, many years ago, one of my PhD students faced a similar problem in statistical thermodynamics. No closed form was found even using special functions and the values were just tabulated.

Rewriting Sangchul Lee's last expression as $$S(b)=\frac 1 e +\sum_{p=1}^\infty \left(e^{-b^p}+e^{-b^{-p}}-1\right)$$ and computing for successive values of $b$ here are some results $$\left( \begin{array}{cc} 2 & -0.33274738243290 \\ 3 & -0.02563211829889 \\ 4 & +0.08370527009108 \\ 5 & +0.14417082789722 \\ 6 & +0.18389984613125 \\ 7 & +0.21207087052753 \\ 8 & +0.23297803691263 \\ 9 & +0.24903012631702 \\ 10 & +0.26170148665894 \\ 11 & +0.27194029136964 \\ 12 & +0.28037846795977 \\ 13 & +0.28745009983337 \\ 14 & +0.29346160695070 \\ 15 & +0.29863473335839 \\ 16 & +0.30313359924031 \\ 17 & +0.30708215624204 \\ 18 & +0.31057572429427 \\ 19 & +0.31368879397943 \\ 20 & +0.31648041901516 \end{array} \right)$$