Chapter 6: Problem 7
Verify \((6-76)\)
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Chapter 6: Problem 7
Verify \((6-76)\)
These are the key concepts you need to understand to accurately answer the question.
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Let \(F(x)=\sum_{1}^{\infty} x^{2} /\left(x^{2}+n^{2}\right)\). Study the uniform convergence of this series, and investigate the existence of \(\lim _{x+0} F(x), \lim _{x \downarrow 0} F(x) / x, \lim _{x \downarrow 0} F(x) / x^{2}, \lim _{x \uparrow \infty} F(x), \lim _{x+\infty} F(x) / x^{2}\), \(\lim _{x \uparrow \infty} F(x) / x\)
If \(f(x)=a_{1} x+a_{2} x^{2}+\cdots\), then \(1 / f(x)=1-a_{1} x+\left\\{\left(a_{1}\right)^{2}-a_{2}\right\\} x^{2}+\left\\{2 a_{1} a_{2}-\left(a_{1}\right)^{3}-a_{3}\right\\} x^{3}\) \(+A x^{4}+\cdots\) Find \(A\)
Evaluate \(\int_{0}^{\infty} \frac{1-\cos x}{x^{2}} \cdot d x\)
In terms of the gamma function, evaluate the following integrals: (a) \(\int_{0}^{1} \frac{x^{3} d x}{\sqrt{1-x^{3}}}\) (b) \(\int_{0}^{1} \frac{d x}{\sqrt{x \log (1 / x)}}\) (c) \(\int_{0}^{1}[1-1 / x]^{1 / 3} d x\) (d) \(\int_{0}^{\pi / 2} \sqrt{\tan \theta} d \theta\)
Investigate the existence and uniformity of the limit
$$
\lim _{t \rightarrow 0} \frac{e^{x t}-1}{x} \quad 0
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