Asymptotic expansion of the sequence $u_n=\sum\limits_{k=1}^{n}e^{-b^{-k}}$

Question

Let $\left(u_n\right)_{n\in\mathbb{N^*}}$ be the sequence defined $\forall n\in\mathbb{N^*}$ by : $$u_n=\sum\limits_{k=1}^{n}e^{-b^{-k}}$$ With $b\in(1,+\infty)$.

We can instictively say that when $n\to\infty$, since $\lim\limits_{k\to\infty}e^{-b^{-k}}\to 1$, it will tend to be like $\sum\limits_{k=1}^{n}1=n$, and thus I expect the dominant term of the asymptotic expansion of $\left(u_n\right)$ to be $n$ :

$$u_n=n+o(n)$$

And numerical simulation shows indeed that $u_n\sim n$ for large n.

Now, this is no demonstration ; and not only am I trying to prove it rigorously, but also (and mainly) to expand it a bit more and find out the next few terms (well, at least the second term) of the asymptotic expansion...

Any suggestion ?

Answer 1

Since $e^{-x}=1-x+x^2/2!-x^3/3!+\ldots$, your sum is \begin{align} \sum_{k=1}^n e^{-b^{-k}}&= n-\sum_{m=1}^\infty \frac{(-1)^{m+1}}{m!}\frac{1-b^{-mn}}{b^m-1} \\&=n-\frac{1-b^{-n}}{b-1}+\frac{1}{2}\frac{1-b^{-2n}}{b^2-1}-\frac{1}{6}\frac{1-b^{-3n}}{b^3-1}\ldots+\frac{(-1)^{m+1}}{m!}\frac{1-b^{-mn}}{b^m-1}+\ldots \end{align} Immediately, this tells us that $u_n\sim n-\sum_{m=1}^\infty \frac{(-1)^{m+1}}{m!}\frac{1}{b^m-1}$. If $b$ is large enough, then $u_n\approx_{n\to\infty} n+e^{-b^{-1}}-1\sim n -\frac{1}{b}$.

Answer 2

If we use the Taylor series expansion of $e^x$: $$e^x=\sum_{n=0}^{\infty}\frac{x^n}{n!}$$ We have the following asymptotic estimation for your sequence (as for large $n$ we have $n \sim \infty$): $$u_n \sim \sum_{k=1}^{\infty}\sum_{n=0}^{\infty} \frac{(-b^{-k})^n}{n!}$$ $$=\lim_{n \to \infty}\sum_{k=1}^{n}\Big(1-\frac{1}{b^k}+\frac{1}{2! \cdot b^{2k}}-\frac{1}{3! \cdot b^{3k}}+...\Big)$$ $$= n-\frac{1}{b-1}+\frac{1}{2! \cdot (b^2-1)} - \frac{1}{3! \cdot (b^3-1)}+...$$ $$=n+\sum_{k=1}^{\infty}\frac{(-1)^k}{k! \cdot(b^k-1)}$$ In the above expansion I used the fact that $1+x+x^2+...=\frac{1}{1-x}$ which is valid as $|\frac{1}{b^k}| \lt 1$ for the given range of $b$.

Answer 3

We have \begin{align*} u_n &= \sum_{k=1}^{n} \sum_{r=0}^\infty \frac{(-b^{-k})^r}{r!} \\ &= \sum_{r=0}^\infty \sum_{k=1}^{n} \frac{(-b^{-k})^r}{r!} \\ &= \sum_{r=0}^\infty \frac{(-1)^{r+1}}{r!}\frac{b^{-nr}-1}{b^r-1} \\ &= n + \frac{b^{-n}-1}{b-1} + \cdots \end{align*}