For what values of M does this integral converge?

  • Context: Graduate 
  • Thread starter Thread starter nonaa
  • Start date Start date
  • Tags Tags
    Integral
Click For Summary
SUMMARY

The integral \int_{-\infty}^{0}\frac{x.e^x}{(1-e^{2x})^M}dx converges for specific values of M. The analysis indicates that for convergence, M must satisfy the condition M < 2. The discussion highlights the importance of understanding the behavior of the integral as x approaches both 0 and negative infinity, emphasizing that the integral must remain finite across its entire range.

PREREQUISITES
  • Understanding of improper integrals
  • Familiarity with limits and convergence criteria
  • Knowledge of exponential functions and their properties
  • Basic calculus skills, particularly integration techniques
NEXT STEPS
  • Study the convergence of improper integrals in detail
  • Learn about the behavior of exponential functions as limits approach infinity
  • Explore techniques for evaluating integrals involving exponential decay
  • Investigate the application of the Dominated Convergence Theorem
USEFUL FOR

Mathematicians, calculus students, and anyone interested in advanced integration techniques and convergence analysis of improper integrals.

nonaa
Messages
17
Reaction score
0
For what values of M does this integral converge?

[tex]\int_{-\infty}^{0}\frac{x.e^x}{(1-e^{2x})^M}dx[/tex]


My pathetic attempt for solution ended here:

[tex]\frac{2x.e^x}{2.(-1)^M(e^{2x}-1)^M}.x^{M-1}\stackrel{x\rightarrow0}{\longrightarrow}\frac{(-1)^M}{2}[/tex]

[tex]M-1<1, M<2[/tex]

Am I going in the right direction? And what to do if [tex]x\rightarrow\infty[/tex]?
 
Physics news on Phys.org
Welcome to PF!

Hi nonaa! Welcome to PF! :smile:

No, you're misreading the question.

It means for what values of M is the whole integral finite. :wink:
 


Hi, I'm here for almost a day but you people are so nice and helpful :) I'm glad I've found this forum :)

And how to do this? I've never solved such type of problems before. Would you give me a little hint?
 

Similar threads

High School Straightforward integration…
  • · Replies 27 ·
Replies
27
Views
3K
Undergrad Derive the orthonormality condition for Legendre polynomials
  • · Replies 30 ·
2
Replies
30
Views
2K
Undergrad Why does the integral of sine of x^2 from - infinity to + infinity diverge?
  • · Replies 4 ·
Replies
4
Views
3K
Graduate Challenging integral involving exponentials and logarithms
  • · Replies 14 ·
Replies
14
Views
3K
Undergrad On convolution theorem of Laplace transform: Schiff
  • · Replies 4 ·
Replies
4
Views
3K
Undergrad Verifying Integration of ##\int_0^1 x^m \ln x \, \mathrm{d}x##
  • · Replies 1 ·
Replies
1
Views
3K
Graduate Solving a power series
  • · Replies 3 ·
Replies
3
Views
4K
Graduate Multiplying divergent integrals using Hardy fields approach
  • · Replies 3 ·
Replies
3
Views
2K
MHB Indefinite integral in division form
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Different Substitution for Solving an Integral
  • · Replies 6 ·
Replies
6
Views
3K