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

Which of the following series converge?

I. $\sum _{n=1}^{\infty }\frac{{\left(n+3\right)}^2}{n\left(n+1\right)\left(n+4\right)}$

II. $\sum _{n=1}^{\infty }\left(\frac{3^n}{n!}\right)$

III. $\sum _{n=1}^{\infty }\frac{\left(n+1\right)!}{n^{50}}$

Let $\sum _{n=1}^{\infty }{a}_n$ be a convergent series, with $a_n>0$ for all $n$. Which of the following must be true?

Which of the following series diverge?

I. $\sum _{n=3}^{\infty }\frac{{\left(-1\right)}^{n+1}}{n}$

II. $\sum _{n=1}^{\infty }\left(\frac{\sqrt{n+1}}{\sqrt{n}+1}\right)$

III. $\sum _{n=2}^{\infty }\frac{3}{n^2+2}$

For a particular value of $q$, $\underset{c\to \infty }{\lim }{\int }_1^c\frac{1}{x^q}dx$ diverges to $+\infty$. Which of the following must also be true?

The infinite series $\sum _{n=1}^{\infty }\frac{{\left(-1\right)}^{n+1}}{n^2-n+20}$ converges to a finite sum $S$. What is the minimum number of terms of the series that you would need to use to find an approximation for $S$ that the alternating series error bound guarantees is not more than $\frac{1}{400}$ away from the true value?

Which of the following statements about the series $\sum _{n=1}^{\infty }\frac{\cos \left(n\pi \right)}{\sqrt{2n+7}}$ is true?

Which of the following series converge?

I. $\sum _{n=1}^{\infty }\frac{1}{2n}{\left(\frac{5}{3}\right)}^n$

II. $\sum _{n=1}^{\infty }{\left(-1\right)}^{n+1}\frac{1}{3n+4}$

III. $\sum _{n=2}^{\infty }\frac{1}{n\ln n}$

Which of the following series converge absolutely?

I. $\sum _{n=5}^{\infty }\frac{{\left(-1\right)}^n}{\sqrt{n}-2}$

II. $\sum _{n=1}^{\infty }\frac{\cos \left(2n\pi \right)}{2\pi \left(n!\right)}$

III. $1+1-2+\frac{1}{2}+\frac{1}{2}-1+\frac{1}{3}+\frac{1}{3}-\frac{2}{3}+\frac{1}{4}+\frac{1}{4}-\frac{1}{2}+...$

Let $f$ be a continuous function that is decreasing and positive on $x\ge 1$. A part of the graph of $f$ is shown in the diagram above. For all $n\ge 1$, the series $\sum _{n=1}^{\infty }{a}_n$ has its nth term defined by $a_n=f\left(n\right)$. If $\underset{c\to \infty }{\lim }{\int }_1^{c}f\left(x\right)dx=3$, which of the following could be true?

A function $f$ is defined in power series form as $f\left(x\right)=\sum _{n=1}^{\infty }\frac{x^n}{n^2+n+298}$, which converges for $\left|x\right|\le 1$. Let $s_n\left(x\right)$ be the nth partial sum of the power series. What is the smallest number $n$ for which the alternating series error bound guarantees that $\left|f\left(-1\right)-s_n\left(-1\right)\right|\le 0.0001$?