Introduction to Infinite Series (College Board AP® Calculus BC): Multiple Choice Questions

For what values of $q$ will the series $\sum _{n=1}^{\infty }\frac{1}{n^{4q}}$ and $\sum _{n=1}^{\infty }{\left(\frac{q}{4}\right)}^n$ both converge?

The sum of the infinite geometric series $\frac{7}{2}+\frac{21}{16}+\frac{63}{128}+\frac{189}{1,024}+...$ is

Which of the following series are convergent?

I. $1+\frac{1}{2^3}+\frac{1}{3^3}+...+\frac{1}{n^3}+...$

II. $1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{n}+...$

III. $1-\frac{1}{2}+\frac{1}{2^2}-...+\frac{{\left(-1\right)}^{n+1}}{2^{n-1}}+...$

For what values of $x$ does the infinite series $1+2^x+3^x+...+n^x+...$ converge?

The sum of the infinite geometric series $\sum _{n=1}^{\infty }4·{\left(-\frac{3}{5}\right)}^n$ is

For what values of $q$ will the series $\sum _{n=1}^{\infty }\frac{3}{\left(n^{\frac{q}{5}}\right)}$ and $\sum _{n=0}^{\infty }{\left(\frac{3q}{5}\right)}^n$ both converge?

A function $f$ is defined in power series form as $f\left(x\right)=\sum _{n=0}^{\infty }{\left(-\frac{x}{5}\right)}^n$. What is the value of $f\left(2\right)$?

For what integer $k$, where $k>4$, will the series $\sum _{n=1}^{\infty }{\left(\frac{k}{7}\right)}^n$ and $\sum _{n=1}^{\infty }\frac{{\left(-1\right)}^{kn}}{n}$ both converge?