Infinite Series Convergence Quiz

The integral test states that if $f(x)$ is a non-negative, monotone decreasing, integrable function such that $f(n) = a_n$ for all $n$, then the infinite series $\sum_{n=1}^{\infty} a_n$ and the improper integral $\int_{1}^{\infty} f(x) , dx$ either both converge or both diverge.

The statement of the integral test is that for a non-negative, monotone decreasing, integrable function $f(x)$ such that $f(n) = a_n$ for all $n$, the infinite series $\sum_{n=1}^{\infty} a_n$ and the improper integral $\int_{1}^{\infty} f(x) , dx$ either both converge or both diverge.

The condition for the integral test is that the function $f(x)$ must be non-negative, monotone decreasing, and integrable, and it should satisfy the property $f(n) = a_n$ for all $n$.

<p>The integral test is used to determine the convergence or divergence of an infinite series by comparing it to the convergence or divergence of an associated improper integral.</p> Signup and view all the answers

<p>The integral test establishes a relationship between the convergence of an infinite series and the convergence of an associated improper integral, stating that they either both converge or both diverge.</p> Signup and view all the answers

<p>The integral test states that if $f(x)$ is a non-negative, monotone decreasing, and integrable function such that $f(n) = a_n$ for all $n$, then the infinite series $\sum_{n=1}^{\infty} a_n$ converges if and only if the improper integral $\int_{1}^{\infty} f(x) , dx$ converges.</p> Signup and view all the answers

<p>The integral test applies when $a_n = f(n)$, $f(x)$ is non-negative, monotone decreasing, and integrable, and the implication is that the series $\sum_{n=1}^{\infty} a_n$ converges if and only if the improper integral $\int_{1}^{\infty} f(x) , dx$ converges.</p> Signup and view all the answers

<p>The integral test states that the infinite series $\sum_{n=1}^{\infty} a_n$ converges if and only if the improper integral $\int_{1}^{\infty} f(x) , dx$ converges, where $a_n = f(n)$ and $f(x)$ is non-negative, monotone decreasing, and integrable.</p> Signup and view all the answers

<p>The integral test applies when $a_n = f(n)$, $f(x)$ is non-negative, monotone decreasing, and integrable. The implication is that the series $\sum_{n=1}^{\infty} a_n$ converges if and only if the improper integral $\int_{1}^{\infty} f(x) , dx$ converges.</p> Signup and view all the answers

<p>The conditions for the integral test ensure that the function $f(x)$ meets specific criteria for the test to be valid. The test's statement is that the series $\sum_{n=1}^{\infty} a_n$ converges if and only if the improper integral $\int_{1}^{\infty} f(x) , dx$ converges, where $a_n = f(n)$ and $f(x)$ is non-negative, monotone decreasing, and integrable.</p> Signup and view all the answers