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I have to analyze the convergence of

$$\int _{0}^{+\infty} \frac{\cos \left( x\right) -1} {x^{5 / 2}+5x^{3}}\,dx$$

I've rewritten the integral as

$$ \int _{0}^{+\infty} \frac{\cos \left( x\right) -1} {x^{5 / 2}+5x^{3}}\,dx = \int _{0}^{1} \frac{\cos \left( x\right) -1} {x^{5 / 2}+5x^{3}}\,dx + \int _{1}^{+\infty} \frac{\cos \left( x\right) -1} {x^{5 / 2}+5x^{3}}\,dx $$

For the first part, I can write:

$$ \int _{0}^{1} \frac{\cos \left( x\right) -1} {x^{5 / 2}+5x^{3}}\,dx = \lim_{a \to 0^{+}} \int _{a}^{1} \frac{\cos(x) - 1}{x^{5/2} + 5x^3}\, dx $$

Then, I search for function $g(x)$ so that $ 0 \leq f(x) \leq g(x)$ for $(0,1]$ in order to use convergence criteria. Then, I narrow the expresion in the integral to find something bigger, but that it converges. I make the following steps:

$$ \frac{||\cos(x) - 1||}{||x^{5/2} + 5x^3||} \leq \frac{2}{||x^{5/2} + 5x^x||} \leq \frac{2}{||x^{5/2}||} \leq \frac{2}{\sqrt{x}} $$

knowing that $0 < x \leq 1$ .

So, $g(x)$ converge because $\int_0^1\frac{1}{x^p}$, with $p < 1$ converges. Then, by the comparison criteria,

$$ \int _{0}^{1} \frac{\cos \left( x\right) -1} {x^{5 / 2}+5x^{3}}\,dx $$

also converges.

Is the reasoning actually correct?

I know I still have to check the second part of the integral, but is more of the same procedure. I just want to check if I'm in the good way!

Thanks a lot for your help!

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(1) Don't use the symbol of norm when you actually mean absolute value; (2) $\,x>0\Longrightarrow x^{5/2}>x^{1/2}\Longleftrightarrow x^5>x\Longleftrightarrow x>1\,$ , what makes your last inequality in the fourth row of equations wrong. –  DonAntonio Dec 12 '12 at 3:03
I can see your point... so how would you reduce the last expression, in order to have $p < 1$, if such $p$ exists? –  pmartelletti Dec 12 '12 at 3:16
because if i cannot find a function g that converge, i cannot say anything for f (by these, i mean that if g diverge, it does not mean the f also does... ). I'm ok? –  pmartelletti Dec 12 '12 at 3:33

1 Answer 1

Unfortunately, we have $x^{-5/2}\gt x^{-1/2}$ if $0\lt x\lt 1$, so the approach to treat the first integral does not work. However, the bound $$\left|\frac{\cos \left( x\right) -1} {x^{5 / 2}+5x^{3}}\right|\leqslant \frac 2{x^{5/2} } $$ is sufficient to deduce the convergence of the second integral.

Back to the first one. Since $$\lim_{x\to 0}\frac{\cos x-1}{x^2}=l$$ for some positive and finite constant $l$, the convergence of $\int_0^ 1 (\cos x-1)/ (x^{5/2}+5x^3)\mathrm dx$ is equivalent to that of $\int_0^1\mathrm dx/ (x^{1/2}+5x )$, which is convergent since for $x\in(0,1)$, $$0\leqslant \frac 1{x^{1/2}+5x }\leqslant \frac 1{x^{1/2}}. $$

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