# Does the limit $e^{H_{n+1}}$-$e^{H_n}$ exists when $n\to\infty$?

$$\lim_{n\to\infty}{e^{1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n+1}}-e^{1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}}}$$

I see that it's an indeterminate form ($\infty-\infty$). I tried to factor and I got: $$\lim_{n\to\infty}{\frac{e^{\frac{2}{n+1}}-1}{e^{\frac{1}{n+1}}+1}\cdot e^{1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}}}$$

Now it's $0\cdot \infty$. How can we approach this type of problem?

Thank you!

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It may help that $\log (n+1)<H_n < 1+\log n$. –  Antoine Feb 1 at 12:23

$$e^{H_{n+1}}-e^{H_n}=n(e^{1/(n+1)}-1)e^{(H_n-\ln n)}$$
Thank you. I did the computations and I obtained $-e^\gamma$. It's correct? –  Daniel C Feb 1 at 13:44
Just to reinforce Claude's point, the harmonic sum increases with $n$, and $e^x$ is an increasing function, so the answer can't be negative. –  Barry Cipra Feb 1 at 13:49