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Chapter 1: Problem 10

Find the sum for each of the series: a. \(5+\frac{25}{7}+\frac{125}{49}+\frac{625}{343}+\cdots\) b. \(\sum_{n=0}^{\infty} \frac{(-1)^{n} 3}{4^{n}}\) c. \(\sum_{n=2}^{\infty} \frac{2}{5^{n}}\). d. \(\sum_{n=-1}^{\infty}(-1)^{n+1}\left(\frac{e}{\pi}\right)^{n}\). e. \(\sum_{n=0}^{\infty}\left(\frac{5}{2^{n}}+\frac{1}{3^{n}}\right)\). f. \(\sum_{n=1}^{\infty} \frac{3}{n(n+3)}\) g. What is \(0.569 ?\)

Short Answer

Expert verified
a. 35, b. \(\frac{4}{5}\), c. \(\frac{2}{15}\), d. \(\frac{e}{\pi + e}\), e. \(13\frac{1}{2}\), f. \(\frac{11}{6}\), g. 0.569

Step by step solution

01

Solve part (a)

Identify the first term \(a = 5\), and the common ratio \(r = \frac{25}{7} \div 5 = \frac{5}{7}\). Using the formula for the sum of an infinite geometric series gives \[S = \frac{5}{1 - \frac{5}{7}} = 35\]
02

Solve part (b)

Identify the first term \(a = 3\) and the common ratio \(r = \frac{(-1) 3}{4}\div 3 = -\frac{1}{4}\). The sum becomes \[S = \frac{3}{1 - -\frac{1}{4}} = \frac{4}{5}\]
03

Solve part (c)

The first term is \(a = \frac{2}{5^2} = \frac{2}{25}\) and the common ratio is \(r = \frac{2}{5}\). The sum is \[S = \frac{\frac{2}{25}}{1 - \frac{2}{5}} = \frac{2}{15}\]
04

Solve part (d)

The first term is \(a = \frac{e}{\pi}\) and the common ratio is \(r = -\frac{e}{\pi}\). So, \[S = \frac{\frac{e}{\pi}}{1 - -\frac{e}{\pi}} = \frac{e}{\pi + e}\]
05

Solve part (e)

This is a sum of two geometric series. The first has \(a_1 = 5\) and \(r_1 = \frac{1}{2}\), and the second has \(a_2 = 1\) and \(r_2 = \frac{1}{3}\). The sum of the two series is \[S= \frac{5}{1 - \frac{1}{2}} + \frac{1}{1 - \frac{1}{3}} = 10 + \frac{3}{2} = 13\frac{1}{2}\]
06

Solve part (f)

First, split the given fraction into partial fractions \[\frac{3}{n(n+3)} = \frac{1}{n} - \frac{1}{n+3}\] Then add up the first few terms to notice the sequence \[1 - \left(\frac{1}{4} + \frac{1}{4} - \left(\frac{1}{7} + \frac{1}{7}\right)\right)\] A pattern emerges and the sum becomes \[S = 1 + \frac{1}{2} + \frac{1}{3} = \frac{11}{6}\]
07

Solve part (g)

This is not a series, but a decimal, therefore there is no need for computation. Thus, \(0.569 = 0.569\)

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Most popular questions from this chapter

Write the following in terms of logarithms: a. \(\cosh ^{-1} \frac{4}{3}\). b. \(\tanh ^{-1} \frac{1}{2}\). c. \(\sinh ^{-1} 2\).

Solve the following equations for \(x\) : a. \(\cosh (x+\ln 3)=3\) b. \(2 \tanh ^{-1} \frac{x-2}{x-1}=\ln 2\). c. \(\sinh ^{2} x-7 \cosh x+13=0\).

Do the following: a. Write \((\cosh x-\sinh x)^{6}\) in terms of exponentials. b. Prove \(\cosh (x-y)=\cosh x \cosh y-\sinh x \sinh y\) using the exponential forms of the hyperbolic functions. c. Prove \(\cosh 2 x=\cosh ^{2} x+\sinh ^{2} x\) d. If \(\cosh x=\frac{13}{12}\) and \(x<0\), find \(\sinh x\) and \(\tanh x\). e. Find the exact value of \(\sinh (\operatorname{arccosh} 3)\).

Denest the following, if possible. a. \(\sqrt{3-2 \sqrt{2}}\) b. \(\sqrt{1+\sqrt{2}}\) c. \(\sqrt{5+2 \sqrt{6}}\) d. \(\sqrt[3]{\sqrt{5}+2}-\sqrt[3]{\sqrt{5}-2}\) e. Find the roots of \(x^{2}+6 x-4 \sqrt{5}=0\) in simplified form.

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