I was studying sequences and series from a book when I ran into the following problem. The text before the problem indicated that I should be using Ratio test or Root test to solve it, however root test didn't seem to fit well and ratio test was inconclusive (resulted in a ratio of $1$)

$$\sum_{n=1}^\infty\left(\frac{1}{n} - e^{-n^2}\right)$$

Any hints or actual answers would be greatly appreciated. Thanks in advance Also please comment as to why it is off-topic before voting to close it.


You are correct, in this case the ratio test or the root test is inconclusive.

Hint. Note that $$\sum_{n=1}^N\left(\frac{1}{n} - e^{-n^2}\right)=\sum_{n=1}^N\frac{1}{n} - \sum_{n=1}^Ne^{-n^2}$$ Now recall that $\sum_{n=1}^{\infty}\frac{1}{n}$ is divergent and $$0\leq \sum_{n=1}^Ne^{-n^2}\leq \sum_{n=1}^{\infty}e^{-n}=\frac{1}{e-1}.$$ What may we conclude?


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