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*}
Section 3.3 Additional Problems
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)
\(\frac{x}{1+x^2}\)
(c)
\(\arctan \left(x^3\right)\)
(d)
\(\ln\left(2+x\right)\)
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)
\(e^x\text{,}\) \(a=3\)
(c)
\(x^3+2x^2+3\) , \(a=1\)
(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)
\(\displaystyle\int_{x=0}^1\sqrt[3]{1-x^3}\dx{ x}\)
