Introduction to Infinite Series (College Board AP® Calculus BC): Free Response Questions

A sequence $\left\{a_n\right\}$ is defined by $a_n=\frac{{\left(n-1\right)}^2}{9900}$ for $n\ge 1$. The sequence $\left\{s_n\right\}$ is the sequence of partial sums for the associated series $\sum _{n=1}^{\infty }{a}_n$.

Find the value of $s_{101}-s_{99}$.

It can be shown that $\sum _{n=1}^{\infty }\frac{1}{n^2}=\frac{{\pi }^2}{6}$ and $\sum _{n=1}^{\infty }\frac{{\left(-1\right)}^{n+1}}{n}=\ln 2$.

Determine the value of $\sum _{n=1}^{\infty }\frac{{\left(-1\right)}^{n+1}·7n^2-5n}{n^3}$.

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

Determine whether the infinite series $\sum _{n=1}^{\infty }\frac{2}{3^{2n}}$ converges or diverges. If it converges, find the sum of the series.

Consider the number $0.0\dot{4}2\dot{3}=0.0423423423423...$. By first writing the number in the form of a geometric series, find the value of $0.0\dot{4}2\dot{3}$ as a fraction in lowest terms.

Determine whether or not the infinite series $\sum _{n=2}^{\infty }\frac{3^n-2^n}{6^n}$ converges, and if it converges determine its value.

A function $f$ is given in power series form as $f\left(x\right)=\sum _{n=0}^{\infty }{\left(-1\right)}^{n}3x^{2n}$. Find the value of$f\left(\frac{1}{3}\right)$.

A sequence $\left\{a_n\right\}$ is defined by $a_n=\frac{4}{n^2+2n}$ for $n\ge 1$. The sequence $\left\{s_n\right\}$ is the sequence of partial sums for the associated series $\sum _{n=1}^{\infty }{a}_n$.

Use partial fractions to find an expression for $s_n$ in terms of $n$. Explain why $\sum _{n=1}^{\infty }\frac{4}{n^2+2n}$ converges and find its sum.