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# Week 7 Wednesday Problems
## Reading.
Please read sections 11.8 to 11.10. These are about power series and Taylor and Maclaurin series.
This set of problems is long, but worth doing and understanding! Make sure to understand each of these problem types. We can go over some of them in the remaining class times.
Most of these problems are from the text 11.9 and 11.10. So do give them a look.
## Problems.
1. Suppose a function $f(x)$ has its $n$-th derivative at $x=4$ given by $$f^{(n)}(4)=\frac{(-1)^{n}n!}{3^{n}(n+1)}$$
1. What is the Taylor series of $f$ centered at $x=4$?
2. What is the radius of convergence of this Taylor series?
2. Use the definition of Taylor series to find the first four nonzero terms of the series $f(x)$ centered at the given value $a$:
1. $f(x) = xe^{x}, \quad a=0$
2. $\displaystyle f(x)=\frac{1}{1+x}, \quad a=2$
3. $f(x) = \sqrt[3]{x}, \quad a = 8$
4. $\displaystyle f(x) = \sin(x), \quad a=\frac{\pi}{6}$
5. $f(x) = \ln(x), \quad a= 1$
6. $f(x)=\cos^{2}(x), \quad a= 0$
3. Find the Maclaurin series of the following (assume these series exists for the given $f$)
1. $f(x)=(1-x)^{-2}$
2. $f(x)= \cos(x)$
3. $f(x) = 2x^{4}+3x^{3}-2x^{2}+7x+1$
4. $f(x)=\sinh(x)$
4. Use binomial series to find the powerseries for the following, and state their radius of convergence:
1. $\displaystyle f(x)=\sqrt[4]{1-x}$
2. $\displaystyle f(x)=\frac{1}{(2+x)^{3}}$
3. $\displaystyle f(x)=(1-x)^{3/4}$
5. Use known Maclaurin series to write down the Maclaurin series for the following:
1. $f(x)=\arctan(x^{2})$
2. $f(x) = x \cos(2x)$
3. $f(x)=e^{2x}-e^{-4x}$
4. $f(x) = x \cos(\frac{1}{2}x^{2})$
5. $\displaystyle f(x)=\frac{x}{\sqrt{4+x^{2}}}$
6. Evaluate the following indefinite integrals as an infinite series
1. $\displaystyle\int\sqrt{1+x^{3}} dx$
2. $\displaystyle\int x^{2}\sin(x^{2})dx$
3. $\displaystyle \int \frac{\cos(x)-1}{x}$
4. $\displaystyle\int\arctan(x^{2})dx$
7. Evaluate the following limits using series:
1. $\displaystyle\lim_{x\to 0} \frac{x-\ln(1+x)}{x^{2}}$
2. $\displaystyle \lim_{x\to 0} \frac{1-\cos(x)}{1+x-e^{x}}$
3. $\displaystyle\lim_{x\to 0} \frac{\sin(x) - x +\frac{1}{6}x^{3}}{x^{5}}$
4. $\displaystyle\lim_{x\to 0} \frac{\sqrt{1+x}-1-\frac{1}{2}x}{x^{2}}$
5. $\displaystyle\lim_{x\to 0} \frac{x^{3}-3x+3\arctan(x)}{x^{5}}$
8. Use multiplication or division of powerseries to find the first three nonzero terms in the Maclaurin series for each of the following:
1. $f(x)=e^{-x}\cos(x)$
2. $f(x)=\tan(x)$
3. $\displaystyle f(x) = \frac{x}{\sin(x)}$
4. $\displaystyle f(x) = (\arctan(x))^{2}$
5. $f(x)=\sec(x)$
6. $\displaystyle f(x)=e^{x} \ln(1+x)$
7. $f(x)=e^{x}\sin^{2}(x)$
9. Find the elementary function represented by the following powerseries:
1. $\displaystyle \sum_{n=0}^{\infty} (-1)^{n}\frac{x^{4n}}{n!}$
2. $\displaystyle \sum_{n=0}^{\infty} (-1)^{n-1}\frac{x^{4n}}{n}$
3. $\displaystyle\sum_{n=0}^{\infty}(-1)^{n} \frac{x^{2n+1}}{2^{2n+1}(2n+1)}$
4. $\displaystyle\sum_{n=0}^{\infty}(-1)^{n} \frac{x^{2n+1}}{2^{2n+1}(2n+1)!}$
10. Find the sum of the following series. Express in exact expression, not decimal approximations.
1. $\displaystyle\sum_{n=0}^{\infty} \frac{(-1)^{n}}{n!}$
2. $\displaystyle\sum_{n=0}^{\infty} \frac{(-1)^{n}\pi^{2n}}{6^{2n}(2n)!}$
3. $\displaystyle\sum_{n=1}^{\infty}(-1)^{n-1} \frac{3^{n}}{n5^{n}}$
4. $\displaystyle\sum_{n=0}^{\infty} \frac{3^{n}}{5^{n}n!}$
5. $\displaystyle\sum_{n=0}^{\infty} \frac{(-1)^{n} \pi^{2n+1}}{4^{2n+1}(2n+1)!}$
6. $\displaystyle 1-\ln2+ \frac{(\ln 2)^{2}}{2!} - \frac{(\ln 2)^{3}}{3!} + \cdots$
7. $\displaystyle 3 + \frac{9}{2!} + \frac{27}{3!} + \frac{81}{4!}+\cdots$
11. Use powerseries / Maclaurin series to find the specified derivatives at the given point:
1. $\displaystyle f(x)=\frac{x}{1+x^{2}}$, find $f^{(101)}(0)$
2. $f(x) = x \sin(x^2)$, find $f^{(203)}(0)$
3. $f(x) = (1+x^{3})^{30}$, find $f^{(58)}(0)$
12. Approximate the following definite integrals to within the indicated accuracy, you might need either Taylor's inequality or Alternating Series Estimate:
1. $\displaystyle\int_{0}^{1/2} x^{3 }\arctan(x)dx$ to within $0.0001$
2. $\displaystyle\int_{0}^{1} x^{3 }\sin(x^{4})dx$ to within $0.0001$
3. $\displaystyle\int_{0}^{0.4} \sqrt{1+x^4}dx$ to within $0.0001$