I need help using the limit comparison test for $sum frac{1}{sqrt{n^2 + 1}}$

I need to determine whether the following series converges or diverges:

$$sum_{n=1}^{infty} frac{1}{sqrt{n^2 + 1}}$$

I’m having trouble finding a series to compare this to but I was thinking maybe $1/n^3$.

Answer

Hint:

$$(n + 1) ^2 = n^2 + 2n + 1 > n^2 + 1$$

Alternatively we may use the Limit test

$$lim_{nto infty} frac{1/(sqrt{n^2 +1})}{1/n} = lim_{n to infty} frac{n}{sqrt{n^2 + 1}} = color{#f05}1 > 0$$

then $sum frac{1}{sqrt{n^2 + 1}}$ diverges because $sum frac{1}{n}$ diverges.

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