W09 Regular

Due date: Sunday 3/15, 11:59pm

Positive series

01

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) ${\displaystyle \sum _{n=1}^{\infty }\frac{1}{\sqrt[3]{n^2}}}$

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Solution

Solutions---0180-04

02 $★$

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.

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Solution

Solutions---0180-06

03

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: ${\displaystyle \sum _{n=2}^{\infty }\frac{n^2}{n^4-1}}$

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Solution

Solutions---0180-05

Alternating series

04

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

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Solution

Solutions---0190-07

Sequences and series - additional practice

05

06

Limits and convergence

For each sequence, either write the limit value (if it converges), or write ‘diverges’.

(a) ${1.01}^n$ $$ (b) ${\displaystyle 2^{1/n}}$ $$ (c) ${\displaystyle \frac{n!}{9^n}}$ $$ (d) ${\displaystyle \frac{3n^2+n+2}{2n^2-3}}$

(e) ${\displaystyle \frac{\cos n}{n}}$ $$ (f) $\ln 5^n-\ln n!$ $$ (g) ${\displaystyle {(2+\frac{4}{n^2})}^{1/3}}$ $$ (h) ${\displaystyle n\sin \frac{\pi }{n}}$

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Solution

Solutions---0140-06

06 $★$

07

Limits and convergence

For each sequence, either write the limit value (if it converges), or write ‘diverges’.

(a) ${\displaystyle \ln \left(\frac{2n+1}{3n+4}\right)}$ $$ (b) ${\displaystyle \frac{e^n}{2^n}}$ $$ (c) ${\displaystyle \frac{{(\ln n)}^2}{n}}$ $$ (d) ${\displaystyle \frac{{(-1)}^n{(\ln n)}^2}{n}}$

(e) ${\displaystyle \frac{3-4^n}{2+7\cdot 4^n}}$ $$ (f) ${\displaystyle {(1+\frac{1}{n})}^n}$ $$ (g) ${\displaystyle \frac{1}{\ln (1+\frac{1}{n})}}$

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Solution

Solutions---0140-07

07 $★$

03

Geometric series

Compute the following summation values using the sum formula for geometric series.

(a) ${\displaystyle \sum _{n=-4}^{\infty }{(-\frac{4}{9})}^n}$ $$ (b) ${\displaystyle \sum _{n=0}^{\infty }{e}^{3-2n}}$

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Solution

Solutions---0150-03

08 $★$

02

Geometric series

Compute the following summation values using the sum formula for geometric series.

(a) ${\displaystyle \sum _{n=0}^{\infty }{5}^{-n}}$ $$ (b) ${\displaystyle \sum _{n=0}^{\infty }\frac{2+3^n}{5^n}}$

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Solution

Solutions---0150-02

09

04

Repeating digits

Using the geometric series formula, find the fractional forms of these decimal numbers:

(a) $0.\bar{2}=0.222222\dots$ $$ (b) $0.4\bar{9}=0.4999999\dots$

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Solution

Solutions---0150-04

10

07

Total area of infinitely many triangles

Find the area of all the triangles as in the figure:

(The first triangle from the right starts at $1$, and going left they never end.)

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Solution

Solutions---0150-07