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Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

use integration, the Direct Comparison Test, or
the Limit Comparison Test to test the integrals for convergence. If
more than one method applies, use whatever method you prefer.

Expert Solution

Step 1

The given integral $\int_{0}^{\infty} \frac{d θ}{1 + e^{θ}}$ .

We have to find the convergence of the integral.

Step 2

Limit comparison test for integrals:

Suppose $f (x), g (x) > 0$ are positive, continuous functions defined on $[a, b)$ such that

$\lim_{x \to b} \frac{f (x)}{g (x)} = c \neq 0, \infty$ , then $\int_{a}^{b} f (x) d x$ converges exactly when $\int_{a}^{b} g (x) d x$ converges.

Step 3

The given integral $\int_{0}^{\infty} \frac{d θ}{1 + e^{θ}}$ .

Let $f (θ) = \frac{1}{1 + e^{θ}}$ and $g (θ) = \frac{1}{e^{θ}}$ , both functions are positive and continuous on the interval $[0, \infty)$ .

Now,

$\begin{array}{rcl} \lim_{θ \to \infty} \frac{f (θ)}{g (θ)} & = & \lim_{θ \to \infty} \frac{\frac{1}{1 + e^{θ}}}{\frac{1}{e^{θ}}} \\ = & \lim_{θ \to \infty} \frac{e^{θ}}{1 + e^{θ}} \\ = & \lim_{θ \to \infty} \frac{1}{\frac{1}{e^{θ}} + 1} \\ = & \frac{1}{0 + 1} \\ \lim_{θ \to \infty} \frac{f (θ)}{g (θ)} & = & 1 \end{array}$

Here, $\lim_{θ \to \infty} \frac{f (θ)}{g (θ)} = 1 \neq 0, \infty$ .

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