Problems

01

Integral Test (IT)

Use the Integral Test to determine whether the series converges:

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

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

02

Direct Comparison Test (DCT)

Determine whether the series is convergent by using the Direct Comparison Test.

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

(a) ${\displaystyle \sum _{n=1}^{\infty }\frac{1}{n^{1/3}+2^n}}$ $$ (b) ${\displaystyle \sum _{k=2}^{\infty }\frac{\sqrt{k}}{k-1}}$

03

Limit Comparison Test (LCT)

Use the Limit Comparison Test to determine whether the series converges:

$${\displaystyle \sum _{n=1}^{\infty }\frac{1}{\sqrt{n}+\ln n}}$$

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

04

Integral Test (IT)

Determine whether the series is convergent by using the Integral Test.

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

(a) ${\displaystyle \sum _{n=1}^{\infty }\frac{1}{n^{1.1}}}$ $$ (b) ${\displaystyle \sum _{n=1}^{\infty }n{e}^{-n^2}}$ $$ (c) ${\displaystyle \sum _{n=1}^{\infty }\frac{1}{\sqrt[3]{n^2}}}$

05

IT, DCT, LCT

Determine whether the series converges by checking applicability and then applying the designated convergence test.

(a) Integral Test: ${\displaystyle \sum _{n=2}^{\infty }\frac{\ln n}{n^2}}$

(b) Direct Comparison Test: ${\displaystyle \sum _{n=1}^{\infty }\frac{n^3}{n^5+4n+1}}$

(c) Limit Comparison Test: ${\displaystyle \sum _{n=2}^{\infty }\frac{n^2}{n^4-1}}$

06

Limit Comparison Test (LCT)

Use the Limit Comparison Test to determine whether the series converges:

$${\displaystyle \sum _{n=1}^{\infty }\frac{e^n+n}{e^{2n}-n^2}}$$

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