Question

Determine if the following is convergent or divergent. Evaluate if possible

The intergral from 0 to infinite (s*e^5s) ds Determine if the following is convergent or divergent. Evaluate if possible

The intergral from 0 to infinite (s*e^5s) ds @Mathematics

Answer 1

without doing any integration, we can look at the function s*e^(5s) and see that it diverges \[\lim_{s \rightarrow \infty}se^{5s}=\infty\] By definition, "an integral of a divergent function is divergent," meaning that if you are adding up all the points, which is what integration is, towards infinity of points that would equal infinity, the integral diverges.

Answer 2

OOOOOOO ok

Answer 3

it is divergent

Answer 4

Thanks p-4 how would I evaluate the equation please

Answer 5

For most that wouldn't work though, and you would have to set your integral as \[\lim_{a \rightarrow \infty} \int\limits_{0}^{a}s*e^{5s}ds\] and solve (in this case by integration by parts and such), but since we know it is divergent, either you can't evaluate it, or it diverges, or it evaluates to infinity (all of which mean about the same thing).

\[\lim_{a \rightarrow \infty} \int\limits\limits_{0}^{a}s*e^{5s}ds=\lim_{a \rightarrow \infty}[(1/25)(5se^{5s}-e^{5s})]=\lim_{a \rightarrow \infty}(1/25)(5ae^{5a}-e^{5a}+1)\]\[=e^{\lim_{a \rightarrow \infty}\ln ((5a-1)(e^{5a}))}=e^{\lim_{a \rightarrow \infty}(\ln (5a-1))+5a}=e^{\infty}=\infty\]


Yeah, it is a slightly annoying limit to evaluate correctly.

Answer 6

\[\int\limits_{0}^{\infty}s\cdot e^{5s}ds>\int\limits_{1}^{\infty}s\cdot e^{5s}ds>\int\limits_{1}^{\infty}1ds=\infty\]

Answer 7

um... yeah, that works too XD the method i outline though can actually be used for all improper integrals, cause for some it isn't that simple.

Answer 8

but this one is that simple...so why make it harder.