Prove that $b_n$ is a Cauchy's sequence

Question

Let $\{b_n\}$ is a sequence such that $b_1=1, b_{n+1} = b_n + (1+ 1/n)^{-n^2}$. Then prove that $~b_n~$ is a Cauchy's sequence

ok I wanna prove that it converges thus it meets Cauchy's criterion and not the opposite..

Answer 1

You can check that $$b_{n+1}=1+\sum_{k=1}^n \left(1+\frac 1 k\right)^{-k^2}$$ and since $$\begin{split}\left(1+\frac 1 k\right)^{-n^2}&=\exp\left(-k^2\ln\left(1+\frac 1 k\right)\right)\\ &=\exp\left(-k^2\left(\frac 1 k+\mathcal O\left(\frac 1 {k^2}\right)\right)\right)\\ &=\exp\left(-k+\mathcal O\left(1\right)\right)\\ &=\mathcal O(e^{-k}) \end{split}$$ the series converges, and so does the $\{b_n\}$ sequence.

Answer 2

Hint: use the root test on the series $\sum_{k=1}^{\infty}(1+\frac{1}{n})^{-n^2}$