W09 Stepwise

Due date: Thursday 3/12, 11:59pm

Positive series

01

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.

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Solution

Solutions---0180-01

02

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) ${\displaystyle \sum _{k=2}^{\infty }\frac{\sqrt{k}}{k-1}}$

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Solution

Solutions---0180-02

03

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.

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Solution

Solutions---0180-03

Alternating series

04

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

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Solution

Solutions---0190-01

05

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?

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Solution

Solutions---0190-03