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

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May 28, 2022
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- 2.2: Power Series as Infinite Polynomials
- 3: Questions Concerning Power Series

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

$\ln (1 - x^2)$
$\frac{x}{1 + x^2}$
$\arctan (x^3)$
$\ln (2 + x)$ [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 $\sum_{n=0}^{\infty } a_n (x - a)^n$ .

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

Q6

Evaluate the following integrals as series.

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

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