Convergence

[Zengbob]

how would you show the integral from 0 to infinity of the function f(t) converges for all a,b >0 , when the function is f(t) = (e^-at -e^-bt ) / t

 

[Steven G]

This is a math help forum where we help students with their math problems. If you had read the forum's guidelines not only would you have received help by now but you would have known that you need to inform us what type of help you need. Maybe show us your work so we can see where you are getting stuck--show us something.

 

[Zengbob]

Ah apologies for that, well I have attempted to carry out the limit comparison test using the numerator as g(t) seeing as the denominator would only make it smaller. But attempting t take limits after simplifying would leave me with the integral from 0 to infinity of the reciprocal of t which doesn't converge (known as its a std test function.) I believe the limit comparison test is the best way to answer this kind of problem but to ensure it converges for a,b >0, picking a function is difficult

 

[Steven G]

Nope, I read your first sentence and stopped. What you said is not true! Dividing by t will not necessarily make the fraction smaller.

If k>0, consider k/1=k, k/(1/2)=2k>k, k/(1/3)=3k>k, k/(1/4)=4k>k...

I am dividing k by various numbers and the fraction are NOT smaller than k. Sorry. Try again

 

[lex]

Attached is a rather long-winded method. I'm sure someone has a nice 3-line solution.