## Related questions with answers

Question

Determine whether the statement is true and give an explanation or counterexample. The sequence of partial sums associated with the series $\sum_{k=1}^{\infty} \frac{1}{k^2+1}$ converges.

Solution

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1 of 2**h)**

This statement is **true**. If we prove that the observed series converges, we will have that the sequence of partial sums associated with this series also converges. Denote $a_k=\frac{1}{k^2+1}$. If we apply the Comparison test with the comparison series $\sum \frac{1}{k^2}$, we have that the series $\sum a_k$ converges.

The series $\sum \frac{1}{k^2}$ is a convergent series as a $p$-series with $p=2>1$ and the following relation holds:

$a_k=\dfrac{1}{k^2+1}<\dfrac{1}{k^2}.$

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