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\flushleft\underline{Homework 4 --- MATH 1586 Spring 2018} \\
Recall the geometric series: for $|r|<1$, we have $\displaystyle\sum_{k=0}^{\infty} r^k = \dfrac{1}{1-r}$. If $|r|>1$, then $\displaystyle\sum_{k=0}^{\infty} r^k$ diverges.

The ratio test: to see if $\displaystyle\sum_{k=0}^{\infty} a_k$ converges or not, compute $\displaystyle\lim_{k \rightarrow \infty} \left| \dfrac{a_{k+1}}{a_k} \right|$. If that limit is $>1$, the series diverges. If that limit is $<1$, then the series converges. If that limit equals $1$, then the ratio test offers no conclusion. 

The Taylor series (centered at $0$) of a function $f(x)$ is given by
$$f(x) = \displaystyle\sum_{k=0}^{\infty} \dfrac{f^{(k)}(0)}{k!} x^k,$$ where $f^{(k)}$ denotes the $k$th derivative of $f$ (with $k=0$ meaning take no derivative).
\begin{enumerate}
\item Calculate $\displaystyle\sum_{k=1}^4 \dfrac{1}{k^2+2k+1}$
\item Does $\displaystyle\sum_{k=0}^{\infty} 2^k$ converge? If so, find its value. If it does not, why not?
\item Does $\displaystyle\sum_{k=0}^{\infty} \dfrac{1}{(-3)^k}$ converge? If so, find its value. If it does not, why not?
\item The sine function  $\sin$ is defined by the series
$$\sin(t) = \displaystyle\sum_{k=0}^{\infty} \dfrac{(-1)^k t^{2k+1}}{(2k+1)!}.$$
The so-called ``sine integral function" $\mathrm{Si}$ (used in, e.g. signal processing) is defined by the formula
$$\mathrm{Si}(x)=\displaystyle\int_0^x \dfrac{\sin(t)}{t} \mathrm{d}t.$$
Use the Taylor series for $\sin(x)$ and integration term-by-term to find the Taylor series for $\mathrm{Si}(x)$.
\item Use the ratio test to try to conclude whether or not the series $\displaystyle\sum_{k=0}^{\infty} \dfrac{1}{k+1}$ converges.
\item Use the ratio test to try to conclude whether or not the series $\displaystyle\sum_{k=0}^{\infty} \dfrac{1}{k 2^k}$ converges.
\item Use the ratio test to try to conclude whether or not the series $\displaystyle\sum_{k=0}^{\infty} \dfrac{k!}{2^k}$ converges.
\item Use the definition of Taylor series to compute the first three nonzero terms of the Taylor series for $f(x)=e^x \ln(x^2+1)$. 
\end{enumerate}

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