Does the power series $\sum_{n=1}(1 - 1/sqrt(n))^n $ converge or diverge?
I tried root test but it doesn't work. Which convergence test should I try to solve the problem?
2026-04-17 21:26:36.1776461196
Convergence of the power series
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Outline: After a while, $\left(1-\frac{1}{\sqrt{n}}\right)^{\sqrt{n}}$ will be close to $e^{-1}$, and in particular will be positive and less than $\frac{1}{2}$.
Thus after a while $\left(1-\frac{1}{\sqrt{n}}\right)^{n}$ is positive and less than $\frac{1}{2^{\sqrt{n}}}$.
To finish, show that after a while $2^{\sqrt{n}}\gt n^2$. From this we can conclude that our series converges.