Calculus/Sequences and Series/Exercises

The following exercises test your understanding of infinite sequences and series. You may want to review that material before trying these problems.

Each question is followed by a "Hint" (usually a quick indication of the most efficient way to work the problem), the "Answer only" (what it sounds like), and finally a "Full solution" (showing all the steps required to get to the right answer). These should show up as "collapsed" or "hidden" sections (click on the title to display the contents), but some older web browsers might not be able to display them correctly (i.e., showing the content when it should be hidden). If this is true for your browser (or if you're looking at a printed version), you should take care not to "see too much" before you start thinking of how to work each problem.

Sequences

Consider the infinite sequence

$a_n=n/2^n,n=1,\dots ,\mathrm{\infty }.$

• Is the sequence monotonically increasing or decreasing?

• Is the sequence bounded from below, from above, both, or neither?

• Does the sequence converge or diverge?

Partial sums

Assume that the n^th partial sum of a series is given by $s_n=2-\frac{1}{3^n}.$

• Does the series converge? If so, to what value?

• What is the formula for the n^th term of the series?

Sums of infinite series

Find the value to which each of the following series converges.

• $\sum _{n=0}^{\mathrm{\infty }}\frac{3}{4^n}$

• $\sum _{n=1}^{\mathrm{\infty }}{\left(\frac{2}{e}\right)}^n$

• $\sum _{n=2}^{\mathrm{\infty }}\frac{1}{n^2-n}$

• $\sum _{n=1}^{\mathrm{\infty }}\frac{(-1{)}^n{2}^{n-1}}{3^n}$

Convergence and divergence of infinite series

Determine whether each of the following series converges or diverges. (Note: Each "Hint" gives the convergence/divergence test required to draw a conclusion.)

• $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^2}$

• $\sum _{n=0}^{\mathrm{\infty }}\frac{1}{2^n}$

• $\sum _{n=1}^{\mathrm{\infty }}\frac{n}{n^2+1}$

• $\sum _{n=2}^{\mathrm{\infty }}\frac{1}{\ln n}$

• $\sum _{n=0}^{\mathrm{\infty }}\frac{n!}{2^n}$

• $\sum _{n=1}^{\mathrm{\infty }}\frac{\cos \pi n}{n}$

• $\sum _{n=2}^{\mathrm{\infty }}\frac{(-1{)}^n}{n\ln n-1}$

Absolute and conditional convergence

Determine whether each of the following series converges conditionally, converges absolutely, or diverges. (Note: Each "Hint" gives the test or tests that most easily lead to the final conclusion.)

• $\sum _{n=1}^{\mathrm{\infty }}\frac{(-1{)}^n}{\sqrt{n}}$

• $\sum _{n=2}^{\mathrm{\infty }}\frac{(-1{)}^n\ln n}{n}$

• $\sum _{n=2}^{\mathrm{\infty }}\frac{(-1{)}^{n}n}{(\ln n{)}^2}$

• $\sum _{n=1}^{\mathrm{\infty }}\frac{(-1{)}^n{2}^n}{e^n-1}$

• $\sum _{n=1}^{\mathrm{\infty }}\frac{(-1{)}^n}{{\sin }^{2}n}$

• $\sum _{n=1}^{\mathrm{\infty }}\frac{(-1{)}^{n}n!}{(2n)!}$

• $\sum _{n=1}^{\mathrm{\infty }}\frac{(-1{)}^n{e}^{1/n}}{\arctan n}$

Taylor series

Find the Taylor series for the following functions centered at the given values.

• $f(x)=(1+3x{)}^2,a=1$

• $f(x)=\frac{1}{5-2x},a=2$

• $f(x)={\sin }^{2}3x,a=\frac{\pi }{4}$