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Section 3.3 Additional Problems
Problem 3.3.1 .
Use the geometric series to obtain the series
\begin{align*}
\ln \left(1+x\right)\amp =x-\frac{1}{2}x^2+\frac{1}{3}x^3-\cdots\\
\amp =\sum_{n=0}^\infty\frac{(-1)^n}{n+1}x^{n+1}.{}
\end{align*}
Problem 3.3.2 .
Without using Taylor’s Theorem, represent the following functions as power series expanded about 0 (i.e., in the form
\(\sum_{n=0}^\infty a_nx^n\) ).
(a)
\(\ln\left(1-x^2\right)\)
(b) (c)
\(\arctan \left(x^3\right)\)
(d)
Hint .
\(2+x=2\left(1+\frac{x}{2}\right)\)
Problem 3.3.3 .
Let
\(a\) be a positive real number. Find a power series for
\(a^x\) expanded about 0.
Hint .
\(a^x=e^{\ln\,\left(a^x\right)}\)
Problem 3.3.4 .
Represent the function
\(\) sin
\(x\) as a power series expanded about
\(a\) (i.e., in the form
\(\sum_{n=0}^\infty a_n\left(x-a\right)^n\) ).
Hint .
\(\sin x=\sin \left(a+x-a\right)\text{.}\)
Problem 3.3.5 .
Without using Taylor’s Theorem, represent the following functions as a power series expanded about
\(a\) for the given value of
\(a\) (i.e., in the form
\(\sum_{n=0}^\infty a_n\left(x-a\right)^n\) ).
(a)
\(\ln x\text{,}\) \(a=1\)
(b) (c) (d)
\(\frac{1}{x}\) ,
\(a=5\)
Problem 3.3.6 .
Evaluate the following integrals as series.
(a)
\(\displaystyle\int_{x=0}^1e^{x^2}\dx{ x}\)
(b)
\(\displaystyle\int_{x=0}^1\frac{1}{1+x^4}\dx{ x}\)
(c)
\(\displaystyle\int_{x=0}^1\sqrt[3]{1-x^3}\dx{ x}\)
