Tests for Divergence & Convergence (College Board AP® Calculus BC): Exam Questions

State the necessary conditions for using the integral test to determine whether or not the series $\sum _{n=0}^{\infty }\frac{1}{3^{2n}}$ converges. Use the integral test to show that $\sum _{n=0}^{\infty }\frac{1}{3^{2n}}$ converges.

A function $f$ is defined in power series form by $f\left(x\right)=\sum _{n=0}^{\infty }{\left(-1\right)}^{n}3x^{2n+1}$. Explain whether or not the series will converge for $f\left(1\right)$.

Given that $\sum _{n=0}^{\infty }\frac{1}{e^n}$ is a convergent series, use the limit comparison test to show that $\sum _{n=0}^{\infty }\frac{{\left(-1\right)}^n}{4e^n-3}$ converges absolutely.

Give a value of $p$ such that $\sum _{n=1}^{\infty }\frac{{\left(-1\right)}^n}{n^p}$ converges, but $\sum _{n=1}^{\infty }\frac{1}{n^{3p}}$ diverges. Give reasons why your value of $p$ is correct.

Consider the convergent series $\sum _{n=1}^{\infty }\frac{{\left(-1\right)}^{n+1}n}{\left(2n-1\right)!}=1-\frac{2}{3!}+\frac{3}{5!}-\frac{4}{7!}+...+\frac{{\left(-1\right)}^{n+1}n}{\left(2n-1\right)!}+...$. Show that $\frac{83}{120}$ approximates the value of the series sum with error less than $\frac{1}{1000}$.

The first two terms of the series $g\left(1\right)=\sum _{n=0}^{\infty }\frac{(-1{)}^n}{2e^n+3}$ are used to approximate $g\left(1\right)$. Use the alternating series error bound to determine an upper bound on the error of the approximation.

Determine whether the series $\sum _{n=5}^{\infty }\frac{3}{2n^2-7n+5}$ converges or diverges. State the conditions of the test used for determining convergence or divergence.

The Maclaurin series for a function $f$ is given by $\sum _{n=1}^{\infty }\frac{(n+1)x^n}{n^2{6}^n}$ and converges to $f\left(x\right)$ for all $x$ in the interval of convergence. It can be shown that the Maclaurin series for $f$ has a radius of convergence of $6$.

Determine whether the Maclaurin series for $f$ converges or diverges at $x=6$. Give a reason for your answer.

A function $f$ is given in power series form as $f\left(x\right)=\sum _{n=1}^{\infty }\frac{\left(n+4\right){\left(x-2\right)}^n}{n^2{3}^n}$. Determine whether the series for $f$ converges or diverges at $x=5$. Give a reason for your answer.

Determine whether or not the series $\sum _{n=1}^{\infty }\frac{{\left(-1\right)}^{n+1}\left(3n-2\right)}{2n^2+1}=\frac{1}{3}-\frac{4}{9}+\frac{7}{19}-\frac{10}{33}+...$ converges. Justify your answer.

Use the integral test to prove that the $p$-series $\sum _{n=1}^{\infty }\frac{1}{n^p}$ converges for $p>1$ and diverges for $0<p\le 1$. Be sure to state the necessary conditions for using the integral test to determine convergence or divergence of these series.

Determine whether the series $\sum _{n=6}^{\infty }\frac{19}{6n^2-25n+11}$ converges or diverges. State and confirm the conditions of the test used for determining convergence or divergence.

Determine whether the series $\sum _{n=2}^{\infty }\frac{{\left(-1\right)}^n\ln n}{n}$ converges absolutely, converges conditionally, or diverges. Justify your answer.

The function $g$ has derivatives of all orders for all real numbers. The Maclaurin series for $g$ is given by $g\left(x\right)=\sum _{n=0}^{\infty }\frac{(-1{)}^n{x}^n}{2e^n+3}$ on its interval of convergence.

State the conditions necessary to use the integral test to determine convergence of the series $\sum _{n=0}^{\infty }\frac{1}{e^n}$. Use the integral test to show that $\sum _{n=0}^{\infty }\frac{1}{e^n}$ converges.

Use the limit comparison test with the series $\sum _{n=0}^{\infty }\frac{1}{e^n}$ to show that the series $g\left(1\right)=\sum _{n=0}^{\infty }\frac{(-1{)}^n}{2e^n+3}$ converges absolutely.