## Related questions with answers

Question

State the following. The Integral Test

Solutions

VerifiedSolution A

Solution B

Answered 1 year ago

The Integral Test states that: If $a_n = f(n)$ where $f$ is a continuous, positive, decreasing function on $[1, \infty)$. Then (i) $\sum_{1}^{\infty} a_n$ converges if and only if $\int_{1}^{\infty} f(x) dx$ converges, and (ii) $\sum_{1}^{\infty} a_n$ diverges if and only if $\int_{1}^{\infty} f(x) dx$ diverges.

Answered 1 year ago

Step 1

1 of 2$\textbf{Integral Test}$

Consider the following series

$\sum_{n=k}^{\infty}f(n)$

Where $f(x)$ is positive and monotone decreasing on $\left[k,\infty\right)$.

This series is convergent $\textbf{if and only if}$ the following integral is finite.

$\int_k^{\infty}f(x)\hspace{1mm}dx$

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