Problems

01

Absolute and conditional convergence

Determine whether the series are absolutely convergent, conditionally convergent, or divergent.

Show your work. You must check applicability of tests.

(a) ${\displaystyle \sum _{n=1}^{\infty }\frac{{(-1)}^{n-1}}{n^{1/3}}}$ $$ (b) ${\displaystyle \sum _{n=1}^{\infty }\frac{{(-1)}^n{n}^4}{n^3+1}}$

02

Absolute and conditional convergence

Determine whether the series are absolutely convergent, conditionally convergent, or divergent by applying series tests.

Show your work. You must check that the test is applicable.

(a) ${\displaystyle \sum _{n=1}^{\infty }\frac{{(-1)}^n}{1+\frac{1}{n}}}$ $$ (b) ${\displaystyle \sum _{n=1}^{\infty }\frac{\cos n\pi }{n^3+1}}$

03

Alternating series: error estimation

Find the approximate value of ${\displaystyle \sum _{n=1}^{\infty }\frac{{(-1)}^{n-1}}{n!}}$ such that the error $E_n$ satisfies $|E_n|<0.005$.

How many terms are needed?