Convergence of f_{n} (x) = πxexp(-πx) on (0,∞)

Thread starter aeronautical
Start date Apr 27, 2009
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Sequence

In summary, uniform convergence in a sequence means that the terms of the sequence approach a limit at the same rate, regardless of the value of x. This is different from pointwise convergence, where the terms approach the limit at different rates depending on x. For a sequence to converge uniformly, the limit must exist and the difference between any term and the limit must be able to be made arbitrarily small. A sequence can converge pointwise but not uniformly, and to show that a sequence converges uniformly, various convergence tests such as the Cauchy criterion and the Weierstrass M-test can be used.

Apr 27, 2009

aeronautical

Sequence converge uniformly?

Homework Statement

Define f_{n}: (0,∞) → R, through f_{n} (x) = π*x*exp(-πx), x > 0.
Does the sequence converge uniformly in (0,∞) when n → ∞?

f_{n} = f subscript n

Can somebody please show me all the steps? Any ideas where i can start?

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Apr 27, 2009

aeronautical

Is the Uniform convergence theorem a good approach?

1. What is uniform convergence in a sequence?

Uniform convergence in a sequence means that the terms of the sequence approach a limit at the same rate, regardless of the value of x. In other words, the convergence of the sequence is not dependent on the specific value of x.

2. How is uniform convergence different from pointwise convergence?

In pointwise convergence, the terms of the sequence approach the limit at different rates depending on the value of x. However, in uniform convergence, the terms approach the limit at the same rate regardless of x.

3. What are the conditions for a sequence to converge uniformly?

A sequence converges uniformly if the limit of the sequence exists and the difference between any term in the sequence and the limit can be made arbitrarily small by choosing a large enough value of n.

4. Can a sequence converge pointwise but not uniformly?

Yes, a sequence can converge pointwise but not uniformly. This means that the terms of the sequence approach the limit at different rates depending on x, rather than approaching the limit at the same rate regardless of x.

5. How do we show that a sequence converges uniformly?

To show that a sequence converges uniformly, we must prove that the difference between any term in the sequence and the limit can be made arbitrarily small by choosing a large enough value of n. This can be done by using the definition of uniform convergence and applying various convergence tests such as the Cauchy criterion and the Weierstrass M-test.