how do i check the convergence of $\int^{b}_{a}\frac{dx}{(x-a)^p}$

Question

Given question is to check the convergence of $$\int^{b}_{a}\frac{dx}{(x-a)^p}$$

I have managed to solve it till $$\lim_{\epsilon\to0}\frac{(b-a)^{1-p}}{1-p}-\frac{\epsilon^{1-p}}{1-p}$$

According to value of $p$ how to check convergence and divergence of the integral.

Please Guide. Thank You

You can also try this change: $$t=\frac{1}{x-a}$$ to get: $$\int_{(b-a)^{-1}}^{\infty}{t^{p-2}\mathrm{d}t}$$ and go on after that. — babakks, Sep 03 '17 at 18:17

score 1 · Answer 1 · answered Sep 03 '17 at 16:34

1

Assuming that $a$ and $b$ are constants. If $1-p>0$ when $\epsilon \to 0$, it will cause that fraction to come out as $0$ and your series will be convergent to $$\frac{(b-a)^{1-p}}{1-p}$$

But if $1-p<0$ then $\epsilon^{1-p}\to\infty $ as small numbers like $0.000000000001 $ when exposed to negative power will become very large in value. Hence the limit will not be defined and integral will be divergent

Although, I am not sure about $p=1$

answered Sep 03 '17 at 16:34

The Dead Legend

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For $p=1$ one has to do a separate calculations, getting logarithms. – mickep Sep 03 '17 at 16:37

how do i check the convergence of $\int^{b}_{a}\frac{dx}{(x-a)^p}$

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