## Related questions with answers

Question

Use the Integral Test to determine whether each series converges or diverges. $\sum_{k=1}^{\infty} \frac{1}{k^{2}+1}$

Solution

VerifiedStep 1

1 of 3#### Integral Test

Let $f$ be a continuous, positive, decreasing function on $[k, \infty)$, and let $a_n = f(n)$. Then the following is true:

$\int_k^{\infty} f(x) \ dx \text{ is convergent} \quad \iff \quad \sum_{n = k}^{\infty} a_n \text{ is convergent}$

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