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.