Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Prove that the improper integral $$\int_0^{\infty}\frac{x-\sin x}{x^{7/2}}\ dx$$ diverge or converge.

share|improve this question
    
'^7/2' is asked –  B11b Dec 12 '12 at 18:14
    
You must enclose your formulae between dollar signs otherwise it is almost unreadable. –  DonAntonio Dec 12 '12 at 18:16
    
The result is $4\sqrt{2\pi}/15$ :) –  Hans Engler Dec 12 '12 at 18:58

1 Answer 1

Hints: (1) We have potential troubles at $0$ and because the interval is infinite. Break up the integral into two parts, $0$ to (say) $1$ and $1$ to $\infty$.

(2) Deal with the "$\infty$" part. Should not be hard, the function decreases fast.

(3) For the behaviour near $0$, use the Taylor series $\sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\cdots$. For $x$ near $0$, the top is $\lt \dfrac{x^3}{3}$.

(4) So for positive $x$ near $0$, our function is positive but less than a constant times $\dfrac{1}{x^{1/2}}$.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.