
# 2.E: Calculus in the 17th and 18th Centuries (Exercises)

[ "article:topic", "authorname:eboman" ]

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

### Q1

Use the geometric series to obtain the series

\begin{align*} \ln (1+x) &= x - \frac{1}{2}x^2 + \frac{1}{3}x^3 - \cdots \\ &= \sum_{n=0}^{\infty }\frac{(-1)^n}{n+1}x^{n+1} \end{align*}

### Q2

Without using Taylor’s Theorem, represent the following functions as power series expanded about $$0$$ (i.e., in the form $$\sum_{n=0}^{\infty }a_n x^n$$).

1. $$\ln (1 - x^2)$$
2. $$\frac{x}{1 + x^2}$$
3. $$\arctan (x^3)$$
4. $$\ln (2 + x)$$  [Hint: $$2 + x = 2\left (1 + \frac{x}{2} \right )$$]

### Q3

Let $$a$$ be a positive real number. Find a power series for $$a^x$$ expanded about $$0$$.  [Hint: $$a^x = e^{\ln (a^x)}$$].

### Q4

Represent the function $$\sin x$$ as a power series expanded about a (i.e., in the form $$\sum_{n=0}^{\infty } a_n (x - a)^n$$). n=0 an (x−a)n). [Hint: $$\sin x = \sin (a + x - a)$$].

### Q5

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 $$\sum_{n=0}^{\infty } a_n (x - a)^n$$.

1. $$\ln x, a = 1$$
2.  $$e^x, a = 3$$
3.  $$x^3 + 2x^2 + 3 , a = 1$$
4. $$\frac{1}{x}, a = 5$$

### Q6

Evaluate the following integrals as series.

1. $$\int_{x=0}^{1} e^{x^2}dx$$
2. $$\int_{x=0}^{1} \frac{1}{1 + x^4}dx$$
3. $$\int_{x=0}^{1} \sqrt[3]{1 - x^3}dx$$

### Contributor

• Eugene Boman (Pennsylvania State University) and Robert Rogers (SUNY Fredonia)