Chapter 4: Problem 1
Sum the even numbers between 1000 and 2000 inclusive.
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By using the logarithmic series, prove that if \(a\) and \(b\) are positive and nearly equal then $$ \ln \frac{a}{b} \simeq \frac{2(a-b)}{a+b} $$ Show that the error in this approximation is about \(2(a-b)^{3} /\left[3(a+b)^{3}\right]\).
The \(N+1\) complex numbers \(\omega_{m}\) are given by \(\omega_{m}=\exp (2 \pi i m / N)\) for \(m=\) \(0,1,2, \ldots, N\) (a) Evaluate the following: (i) \(\sum_{m=0}^{N} \omega_{m}\), (ii) \(\sum_{m=0}^{N} \omega_{m}^{2}\), (iii) \(\sum_{m=0}^{N} \omega_{m} x^{m}\). (b) Use these results to evaluate (i) \(\sum_{m=0}^{N}\left[\cos \left(\frac{2 \pi m}{N}\right)-\cos \left(\frac{4 \pi m}{N}\right)\right]\) (ii) \(\sum_{m=0}^{3} 2^{m} \sin \left(\frac{2 \pi m}{3}\right)\).
Determine the positive values of \(x\) for which the following series converges: $$ \sum_{n=1}^{\infty} \frac{x^{n / 2} e^{-n}}{n} $$
Find the limits of the following functions: (a) \(\frac{x^{3}+x^{2}-5 x-2}{2 x^{3}-7 x^{2}+4 x+4}, \quad\) as \(x \rightarrow 0, x \rightarrow \infty\) and \(x \rightarrow 2\); (b) \(\frac{\sin x-x \cosh x}{\sinh x-x}, \quad\) as \(x \rightarrow 0\) (c) \(\int_{x}^{\pi / 2}\left(\frac{y \cos y-\sin y}{y^{2}}\right) d y, \quad\) as \(x \rightarrow 0\).
Determine whether the following series are convergent: (a) \(\sum_{n=1}^{\infty} \frac{n^{1 / 2}}{(n+1)^{1 / 2}}\), (b) \(\sum_{n=1}^{\infty} \frac{n^{2}}{n !}\), (c) \(\sum_{n=1}^{\infty} \frac{(\ln n)^{n}}{n^{n / 2}}\), (d) \(\sum_{n=1}^{\infty} \frac{n^{n}}{n !}\).
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