2.E: Calculus in the 17th and 18th Centuries (Exercises)
- Page ID
- 8274
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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.
- \(\int_{x=0}^{1} e^{x^2}dx\)
- \(\int_{x=0}^{1} \frac{1}{1 + x^4}dx\)
- \(\int_{x=0}^{1} \sqrt[3]{1 - x^3}dx\)