Unit 03 - Essential problems

Sequences

01

L’Hopital practice - converting indeterminate form

By imitating the technique of the L’Hopital’s Rule example, find the limit of the sequence:

$$a_n=\sqrt{n}\ln (1+\frac{1}{n})$$

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Solution

Solutions---0140-01

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

Series basics

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

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

Positive series

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

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

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

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

Ratio test and Root test

02

Ratio and root tests

Apply the ratio test or the root test to determine whether each of the following series is absolutely convergent, conditionally convergent, or divergent.

(a) ${\displaystyle \sum _{n=1}^{\infty }\frac{{(-2)}^n}{n^{100}}}$ $$ (b) ${\displaystyle \sum _{n=0}^{\infty }{\left(\frac{5n}{10n+4}\right)}^n}$ $$ (c) ${\displaystyle \sum _{n=1}^{\infty }\frac{\sqrt{n}}{3^n}}$

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Solution

Solutions---0200-02

Power series: Radius and Interval

02

Power series - radius and interval

Find the radius and interval of convergence for these power series:

(a) ${\displaystyle \sum _{n=0}^{\infty }{(-1)}^n\frac{{(x+3)}^n}{n!}}$ $$ (b) ${\displaystyle \sum _{n=1}^{\infty }{(-1)}^n\frac{{(x-7)}^n}{n}}$ $$ (c) ${\displaystyle \sum _{n=12}^{\infty }{n}^n{(x-2)}^n}$

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Solution

Solutions---0220-02

Series tests: strategy tips

02

Various limits, Part II

Find the limits. You may use $+\infty$ or $-\infty$ or $\mathrm{DNE}$ as appropriate. Braces indicate sequences.

• C = Convergent
• AC = Absolutely Convergent
• CC = Conditionally Convergent
• D = Divergent

$a_n$	${\lim }_{n\to \infty }{a}_n$	$\left\{a_n\right\}$ C or D	${\lim }_{n\to \infty }{(-1)}^n{a}_n$	$\left\{{(-1)}^n{a}_n\right\}$ C or D	$\sum a_n$ AC, CC, or D	$\sum {(-1)}^n{a}_n$ AC, CC, or D
${\displaystyle \frac{4n!}{2^n}}$
${\displaystyle \frac{(n+2)3^n}{n!}}$
${\displaystyle \frac{4^n}{{(3n)}^n}}$
${\displaystyle \frac{1}{(2n+1)!}}$

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Solution

Solutions---0210-02

Power series as functions

01

Modifying geometric power series

Consider the geometric power series $\frac{1}{1-x}=1+x+x^2+x^3+\cdots ={\sum }_{n=0}^{\infty }{x}^n$ for $|x|<1$.

For this problem, you should modify the series for $\frac{1}{1-x}$.

(a) Write ${\displaystyle \frac{1}{5-x}}$ as a power series and determine its interval of convergence.

(b) Write ${\displaystyle \frac{1}{16+2x^3}}$ as a power series and determine its interval of convergence.

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Solution

Solutions---0230-01

03

Finding a power series

Find a power series representation for these functions:

(a) $f(x)={\displaystyle \frac{x^2}{x^4+81}}$ $$ (b) $g(x)=x^2\ln (1+x)$

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Solution

Solutions---0230-03

Taylor and Maclaurin series

01

Maclaurin series

For each of these functions, find the Maclaurin series, and the interval on which the expansion is valid.

(a) $x\ln (1-5x)$ $$ (b) $x^2\cos (x^3)$

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Solution

Solutions---0240-01

05

Discovering the function from its Maclaurin series

For each of these series, identify the function of which it is the Maclaurin series, and evaluate the function at an appropriate choice of $x$ to find the total sum for the series.

(a) ${\displaystyle \sum _{n=0}^{\infty }{(-1)}^n\frac{{\pi }^{2n+1}}{4^{2n+1}(2n+1)!}}$ $$ (b) ${\displaystyle \sum _{n=0}^{\infty }\frac{2^{2n}}{n!}}$ $$ (c) ${\displaystyle \sum _{n=0}^{\infty }{(-1)}^n\frac{{\pi }^{2n+2}}{3^{2n+1}(2n)!}}$

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Solution

Solutions---0240-05

06

Summing a Maclaurin series by guessing its function

For each of these series, identify the function of which it is the Maclaurin series:

(a) ${\displaystyle \sum _{n=0}^{\infty }{(-1)}^n\frac{5x^{4n+2}}{(2n+1)!}}$ $$ (b) ${\displaystyle \sum _{n=0}^{\infty }\frac{{(-5x)}^{n+1}}{n+1}}$

Now find the total sums for these series:

(c) ${\displaystyle \sum _{n=0}^{\infty }\frac{{(-5)}^n}{n!}}$ $$ (d) ${\displaystyle \sum _{n=0}^{\infty }\frac{{(-1)}^n{\pi }^{2n}}{9^n(2n)!}}$

(Hint: for (c)-(d), do the process in (a)-(b), then evaluate the resulting function somewhere.)

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Solution

Solutions---0240-06

09

Large derivative at $x=0$ using pattern of Maclaurin series

Consider the function $f(x)=x^2\sin \left(5x^3\right)$. Find the value of $f^{(35)}(0)$.

(Hint: find the rule for coefficients of the Maclaurin series of $f(x)$ and then plug in $0$.)

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Solution

Solutions---0240-09

Applications of Taylor series

01

Approximating $1/e$

Using the series representation of $e^x$, show that:

$$\frac{1}{e}=\frac{1}{2!}-\frac{1}{3!}+\frac{1}{4!}-\cdots$$

Now use the alternating series error bound to approximate $\frac{1}{e}$ to an error within ${10}^{-3}$.

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Solution

Solutions---0250-01

03

Some estimates using series

Find an infinite series representation of ${\displaystyle {\int }_0^1\sin (x^2)dx}$ and then use your series to estimate this integral to within an error of ${10}^{-3}$.

(Use the error bound formula for alternating series.)

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Solution

Solutions---0250-03