## anonymous 4 years ago Integration help!

1. anonymous

$\int\limits_{1}^{\infty}dx/(x^5(e^{1/x}-1))$

2. anonymous

3. anonymous

first year university

4. mathmate

Doesn't look like an analytic solution. It can be solved by series, or numerical. Are you in any of these?

5. anonymous

show that $e^t \ge 1+t$ for $t \ge0$ hence explain briefly why the integral must converge.

6. mathmate

By MacLaurin's expansion, $$e^t$$ = 1+t/1!+t^2/2!+t^3/3! +... => $$e^t \ge$$ 1+t for t$$\gt 0$$ This means that $$e^t-1 \ge$$ 1+t-1 =t for t$$\gt 0$$ or $$1/(e^t-1) \le$$ 1/t for t$$\gt 0$$ therefore I < integral of 1/x^6

7. mathmate

and $\int\limits_{1}^{\infty} \frac{dx}{x^6} \ = \ \frac{1}{5}$

8. anonymous

i dont think we can use MacLaurin's expansion though

9. mathmate

Then you can show that e^t=1 when t=0. and d(e^t)/dt >1 for t>0, therefore e^t-1 >0 for t>0.

10. anonymous

ok thanks