Lebeque dominated convergence thm

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In summary, the conversation discusses solving the question of finding the limit as n approaches infinity for the integral of sin(x/n)/(1+x/n)^n from 0 to infinity. The responder suggests that the solution is correct and recommends using punctuation in future responses.
  • #1
vandanak
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i got this question in my test i have solved it like this
Lim n->∞∫sin(x/n)/(1+x/n)^ndx integral from 0 to infinity and sin(x/n)/(1+x/n)^n<=e^-x as sin(x/n)<=1 and (1+x/n)^n tends to e^x so we can apply dominated convergence thm as all fn(x) will be less than e^-x so taking limit inside as sin(x/n) tends to zero as n tends to infinity so answer should be 0 well I have not written detailed steps but if my questions is not clear please say it
Is this correct or else how we have to solve it
 
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  • #2
Looks all right to me. Suggestion: use punctuation!
 

1. What is the Lebesgue dominated convergence theorem?

The Lebesgue dominated convergence theorem is a fundamental result in measure theory that states that if a sequence of measurable functions converges pointwise to a limit function and is dominated by an integrable function, then the limit function is also integrable and the integral of the sequence approaches the integral of the limit function.

2. Why is the Lebesgue dominated convergence theorem important?

The Lebesgue dominated convergence theorem is important because it allows for the interchange of the limit and integral operations in certain cases, making it a powerful tool in the study of integration and convergence of functions. It also gives conditions under which the integral of a sequence of functions can be approximated by the integrals of its limit function.

3. How is the Lebesgue dominated convergence theorem different from the Monotone convergence theorem?

The Lebesgue dominated convergence theorem is more general than the Monotone convergence theorem, as it does not require the sequence of functions to be increasing or decreasing. Additionally, the Lebesgue dominated convergence theorem only requires the functions to be dominated by an integrable function, while the Monotone convergence theorem requires the functions to be non-negative and increasing.

4. Can the Lebesgue dominated convergence theorem be extended to sequences of functions with complex values?

Yes, the Lebesgue dominated convergence theorem can be extended to sequences of functions with complex values, as long as the integrable function used for domination is also complex-valued. This extension is known as the Lebesgue dominated convergence theorem for complex-valued functions.

5. In what areas of mathematics is the Lebesgue dominated convergence theorem commonly used?

The Lebesgue dominated convergence theorem is commonly used in areas such as real analysis, measure theory, and functional analysis. It is also an important tool in the study of probability and stochastic processes, as well as in the analysis of differential equations and Fourier series.

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