• Support PF! Buy your school textbooks, materials and every day products Here!

Comparison Test: converges or diverges?

  • Thread starter theRukus
  • Start date
  • #1
49
0

Homework Statement


Determine whether or not the improper integral from 0 to infinite of (e^x)/[(e^2x)+4] converges and if it does, find it's definite value.


Homework Equations





The Attempt at a Solution


I missed the lecture on the Comparison Test, so I'm essentially useless.

I assign g(x) = e^x. Let f(x) be the function defined in the question statement. g(x) > f(x) on 0 to infinite, so if g(x) converges then f(x) converges, correct? I then sub in t as the upper limit and evaluate the function as t approaches infinite. This limit cannot be evaluated, so g(x) converges and therefore f(x) converges, right?

Thanks for any help.
 

Answers and Replies

  • #2
150
0
I'm not sure I follow what you're saying, but here's the basic idea behind the comparison test: let a(n) be the series whose convergence you are trying to determine. If you can find some series b(n) such that |b(n)| > |a(n)| for all n (or at least sufficiently large n), and b(n) converges, then a(n) converges (think of it graphically - b(n) squeezes a(n) to zero). Similarly, if |b(n)| < |a(n)| for sufficiently large n, and b(n) diverges, then a(n) diverges.

In this case, you've used as your comparison e^x, which as you've pointed out is always bigger than f(x). However,

[tex]\int_0^{\infty} e^x\ dx[/tex] diverges, so that doesn't do you any good. You are correct, however, that f(x) converges. Try to think of another function larger than f(x). As a hint: what does that 4 in the denominator do for you?
 

Related Threads on Comparison Test: converges or diverges?

Replies
1
Views
1K
Replies
5
Views
10K
  • Last Post
Replies
6
Views
816
Replies
10
Views
2K
  • Last Post
Replies
0
Views
2K
  • Last Post
Replies
2
Views
824
  • Last Post
Replies
8
Views
1K
Replies
15
Views
2K
Replies
5
Views
2K
Replies
13
Views
2K
Top