Basic Technical Mathematics with Calculus

Allyn J. Washington, Richard S. Evans

Chapter 30

Expansion of Functions in Series - all with Video Answers

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Section 1

Infinite Series

Problem 1

Make the given changes in the indicated examples of this section and then solve the given problems.
In Example $3(\mathrm{a}),$ change $5^{n}$ to $0.5^{n} .$ What other changes occur?

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Problem 2

Make the given changes in the indicated examples of this section and then solve the given problems.
In Example 4, change $n=0$ to $n=1 .$ What is the value of $S ?$

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Problem 3

Give the first four terms of the sequences for which $a_{n}$ is given.
$$a_{n}=n^{3}, \quad n=1,2,3, \ldots$$

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Problem 4

Give the first four terms of the sequences for which $a_{n}$ is given.
$$a_{n}=\frac{2^{n+1}}{n !}, n=1,2,3, \dots$$

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Problem 5

Give the first four terms of the sequences for which $a_{n}$ is given.
$$a_{n}=\frac{n}{n+1}, \quad n=0,1,2, \dots$$

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Problem 6

Give the first four terms of the sequences for which $a_{n}$ is given.
$$a_{n}=\frac{n^{2}+1}{2 n+1}, \quad n=0,1,2, \dots$$

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Problem 7

Give (a) the first four terms of the sequence for which $a_{n}$ is given and $(b)$ the first four terms of the infinite series associated with the sequence.
$$a_{n}=\left(\frac{2}{3}\right)^{n}, \quad n=1,2,3, \dots$$

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Problem 8

Give (a) the first four terms of the sequence for which $a_{n}$ is given and $(b)$ the first four terms of the infinite series associated with the sequence.
$$a_{n}=\frac{1}{n}+\frac{1}{n+1}, \quad n=1,2,3, \dots$$

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Problem 9

Give (a) the first four terms of the sequence for which $a_{n}$ is given and $(b)$ the first four terms of the infinite series associated with the sequence.
$$a_{n}=\cos \frac{n \pi}{2}, \quad n=0,1,2, \ldots$$

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Problem 10

Give (a) the first four terms of the sequence for which $a_{n}$ is given and $(b)$ the first four terms of the infinite series associated with the sequence.
$$a_{n}=\frac{n^{n}}{n !}, \quad n=2,3,4, \dots$$

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Problem 11

Find the nth term of the given infinite series for which $n=1,2,3, \dots$
$$\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\dots$$

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Problem 12

Find the nth term of the given infinite series for which $n=1,2,3, \dots$
$$\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\dots$$

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Problem 13

Find the nth term of the given infinite series for which $n=1,2,3, \dots$
$$\frac{1}{2 \times 3}-\frac{1}{3 \times 4}+\frac{1}{4 \times 5}-\frac{1}{5 \times 6}+\cdots$$

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Problem 14

Find the nth term of the given infinite series for which $n=1,2,3, \dots$
$$\frac{1}{\sqrt{2}}-\frac{1}{2}+\frac{\sqrt{2}}{4}-\frac{1}{4}+\cdots$$

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Problem 15

Find the first five partial sums of the given series and determine whether the series appears to be convergent or divergent. If it is convergent, find its approximate sum.
$$1+\frac{1}{8}+\frac{1}{27}+\frac{1}{64}+\frac{1}{125}+\dots$$

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Problem 16

Find the first five partial sums of the given series and determine whether the series appears to be convergent or divergent. If it is convergent, find its approximate sum.
$$1+2+5+10+17+\dots$$

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Problem 17

Find the first five partial sums of the given series and determine whether the series appears to be convergent or divergent. If it is convergent, find its approximate sum.
$$1+\frac{1}{2}+\frac{2}{3}+\frac{3}{4}+\frac{4}{5}+\dots$$

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Problem 18

Find the first five partial sums of the given series and determine whether the series appears to be convergent or divergent. If it is convergent, find its approximate sum.
$$\frac{1}{3}-\frac{1}{9}+\frac{1}{27}-\frac{1}{81}+\frac{1}{243}-\cdots$$

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Problem 19

Find the first five partial sums of the given series and determine whether the series appears to be convergent or divergent. If it is convergent, find its approximate sum.
$$\sum_{n=0}^{\infty} \sqrt{n}$$

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Problem 20

Find the first five partial sums of the given series and determine whether the series appears to be convergent or divergent. If it is convergent, find its approximate sum.
$$\sum_{n=1}^{\infty} \frac{2}{n(n+1)}$$

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Problem 21

Find the first five partial sums of the given series and determine whether the series appears to be convergent or divergent. If it is convergent, find its approximate sum.
$$\sum_{n=1}^{\infty} \frac{2 n+1}{n^{2}(n+1)^{2}}$$

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Problem 22

Find the first five partial sums of the given series and determine whether the series appears to be convergent or divergent. If it is convergent, find its approximate sum.
$$\sum_{n=1}^{\infty} \frac{n^{2}}{2 n^{2}+1}$$

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Problem 23

Find the first five partial sums of the given series and determine whether the series appears to be convergent or divergent. If it is convergent, find its approximate sum.
$$\sum_{n=1}^{\infty} \frac{\sin n}{4^{n}}$$

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Problem 24

Find the first five partial sums of the given series and determine whether the series appears to be convergent or divergent. If it is convergent, find its approximate sum.
$$\sum_{n=3}^{\infty} \frac{\ln n}{e^{n}}$$

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Problem 25

Test each of the given geometric series for convergence or divergence. Find the sum of each series that is convergent.
$$1+2+4+\cdots+2^{n}+\cdots$$

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Problem 26

Test each of the given geometric series for convergence or divergence. Find the sum of each series that is convergent.
$$1+\frac{1}{2}+\frac{1}{4}+\dots+\frac{1}{2^{n}}+\cdots$$

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Problem 27

Test each of the given geometric series for convergence or divergence. Find the sum of each series that is convergent.
$$1-\frac{1}{3}+\frac{1}{9}-\cdots+\left(-\frac{1}{3}\right)^{n}+\cdots$$

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Problem 28

Test each of the given geometric series for convergence or divergence. Find the sum of each series that is convergent.
$$1-\frac{3}{2}+\frac{9}{4}-\cdots+\left(-\frac{3}{2}\right)^{n}+\cdots$$

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Problem 29

Test each of the given geometric series for convergence or divergence. Find the sum of each series that is convergent.
$$10+9+8.1+7.29+6.561+\cdots$$

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Problem 30

Test each of the given geometric series for convergence or divergence. Find the sum of each series that is convergent.
$$4+1+\frac{1}{4}+\frac{1}{16}+\frac{1}{64}+\cdots$$

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Problem 31

Test each of the given geometric series for convergence or divergence. Find the sum of each series that is convergent.
$$512-64+8-1+\frac{1}{8}-\dots$$

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Problem 32

Test each of the given geometric series for convergence or divergence. Find the sum of each series that is convergent.
$$16+12+9+\frac{27}{4}+\frac{81}{16}+\dots$$

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Problem 33

Find the values of $x$ for which the given series converge.
$$\sum_{n=0}^{\infty}(x-4)^{n}$$

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Problem 34

Find the values of $x$ for which the given series converge.
$$\sum_{n=2}^{\infty} \frac{x^{n}}{5^{n}}$$

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Problem 35

Determine the convergence or divergence of the given sequence. If $a_{n}$ is the term of a sequence and $f(x)$ exists for $x \geq 1$ such that $f(n)=a_{n},$ then $\lim f(x)=$L means $a_{n} \rightarrow$Las $n \rightarrow \infty .$ This lets us analyze convergence or divergence by using the equivalent continuous function. Therefore, if applicable, L'Hospital's rule may be used.
$$a_{n}=2+\frac{2}{n}$$

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Problem 36

Determine the convergence or divergence of the given sequence. If $a_{n}$ is the term of a sequence and $f(x)$ exists for $x \geq 1$ such that $f(n)=a_{n},$ then $\lim f(x)=$L means $a_{n} \rightarrow$Las $n \rightarrow \infty .$ This lets us analyze convergence or divergence by using the equivalent continuous function. Therefore, if applicable, L'Hospital's rule may be used.
$$a_{n}=\frac{5 n^{2}-2 n+3}{2 n^{2}+3 n-1}$$

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Problem 37

Determine the convergence or divergence of the given sequence. If $a_{n}$ is the term of a sequence and $f(x)$ exists for $x \geq 1$ such that $f(n)=a_{n},$ then $\lim f(x)=$L means $a_{n} \rightarrow$Las $n \rightarrow \infty .$ This lets us analyze convergence or divergence by using the equivalent continuous function. Therefore, if applicable, L'Hospital's rule may be used.
$$a_{n}=\frac{e^{n}}{n^{3}}$$

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Problem 38

Determine the convergence or divergence of the given sequence. If $a_{n}$ is the term of a sequence and $f(x)$ exists for $x \geq 1$ such that $f(n)=a_{n},$ then $\lim f(x)=$L means $a_{n} \rightarrow$Las $n \rightarrow \infty .$ This lets us analyze convergence or divergence by using the equivalent continuous function. Therefore, if applicable, L'Hospital's rule may be used.
$$a_{n}=\frac{\ln n}{n^{2}}$$

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Problem 39

Solve the given problems as indicated.
The repeating decimal $0.151515 \ldots$ can be expressed as $\frac{15}{100}+\frac{15}{10,000}+\frac{15}{1,000,000}+\cdots .$ Find the sum of this series.

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Problem 40

Solve the given problems as indicated.
Using a calculator, (a) take successive square roots of 0.01 and then
(b) take successive square roots of $100 .$ From these sequences of square roots, state any general conclusions that might be drawn.

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Problem 41

Solve the given problems as indicated.
Referring to Chapter $19,$ we see that the sum of the first $n$ terms of a geometric sequence is $S_{n}=\frac{a_{1}\left(1-r^{n}\right)}{1-r} \quad(r \neq 1)$ where $a_{1}$ is the first term and $r$ is the common ratio. We can visualize the corresponding infinite series by graphing the function $f(x)=a_{1}\left(1-r^{x}\right) /(1-r)(r \neq 1)$ (or using a calculator that can graph a sequence). The graph represents the sequence of partial sums for values where $x=n,$ since $f(n)=S_{n}$ Use a calculator to visualize the first five partial sums of the series $\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\dots$
What value does the infinite series approach? (Remember: Only points for which $x$ is an integer have real meaning.)

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Problem 42

Solve the given problems as indicated.
Write out the first four terms of the series $\sum_{n=1}^{\infty}\left(\frac{1}{n}-\frac{1}{n+1}\right) .$ Then find the first four partial sums. Does this series appear to converge? Why or why not?

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Problem 43

Solve the given problems as indicated.
When two dice are rolled repeatedly, the probability of rolling a sum of 4 before rolling a sum of 7 is given by $\frac{3}{36}\left[1+\frac{27}{36}+\left(\frac{27}{36}\right)^{2}+\left(\frac{27}{36}\right)^{3}+\cdots\right] .$ Find this probability.

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Problem 44

Solve the given problems as indicated.
If an electric discharge is passed through hydrogen gas, a spectrum of isolated parallel lines, called the Balmer series, is formed. See Fig. $30.3 .$ The wavelengths $\lambda$ (in $\mathrm{nm}$ ) of the light for these lines is given by the formula $\frac{1}{\lambda}=1.097 \times 10^{-2}\left(\frac{1}{2^{2}}-\frac{1}{n^{2}}\right) \quad(n=3,4,5, \ldots)$ Find the wavelengths of the first three lines and the shortest wavelength of all the lines of the series.

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Problem 45

Solve the given problems as indicated.
Use geometric series to show that $\sum_{n=0}^{\infty} x^{n}=\frac{1}{1-x}$ for $|x|<1$.

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Problem 46

Solve the given problems as indicated.
Use geometric series to show that $\sum_{n=0}^{\infty}(-1)^{n} x^{n}=\frac{1}{1+x}$ for $|x|<1$.

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Problem 47

Solve the given problems as indicated.
If term $a_{1}$ is given along with a rule to find term $a_{n+1}$ from term
$a_{1},$ the sequence is said to be defined recursively. If $a_{1}=2$ and $a_{n+1}=(n+1) a_{n},$ find the first five terms of the sequence.

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Problem 48

Solve the given problems as indicated.
A sequence is defined recursively (see Exercise 47 ) by $x_{1}=\frac{N}{2}, x_{n+1}=\frac{1}{2}\left(x_{n}+\frac{N}{x_{n}}\right) .$ With $N=10,$ find $x_{6}$ and
compare the value with $\sqrt{10} .$ It can be seen that $\sqrt{N}$ can be approximated using this recursion sequence.

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