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

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)