Uniform convergence of series

In summary, the conversation is about determining whether the series [sum from n=1 to inf] \frac{e^{-nx}}{n^2} on [0, inf) converges uniformly. The suggested methods for solving this problem are the M test or Cauchy's Principle of uniform convergence. The conversation includes the attempt at a solution, which involved trying to convert the exponential into summation form, but no progress was made. The conversation concludes with a suggestion to consider bounding e-nx since nx is positive.
  • #1
Namo
2
0

Homework Statement



Does the following series converge uniformly?

[sum from n=1 to inf] [itex]\frac{e^{-nx}}{n^2} [/itex] on [0, inf)

Homework Equations



I know I need to use the M test or Cauchys Principle of uniform convergence. My tutor suggests using the former if there is uniform convergence & the latter if there isn't.

The Attempt at a Solution



I tried converting the exponential into summation form to see if that would help, but it didn't get me anywhere. I can't really see any easy way to use the M test.

Could anyone point me in the right direction as to start this problem?

Cheers
 
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  • #2
nx is positive, so what's the first thing that comes to mind to bound e-nx?

You almost never want to turn an exponential into its power series form to answer a question like this
 

What is uniform convergence of series?

Uniform convergence of series is a mathematical concept that refers to the convergence of a sequence of functions to a single function in a way that is independent of the input variable. In other words, the function values approach a fixed value as the number of terms in the series increases.

How is uniform convergence different from pointwise convergence?

Uniform convergence and pointwise convergence are two different types of convergence for sequences of functions. Pointwise convergence means that for each input value, the function values approach a fixed value as the number of terms in the series increases. Uniform convergence, on the other hand, means that the function values approach a fixed value uniformly for all input values.

Why is uniform convergence important?

Uniform convergence is important because it guarantees that the limit of a sequence of functions is a continuous function. This is useful in many areas of mathematics and physics, as it allows for the analysis and approximation of complicated functions.

How do you prove uniform convergence of a series?

To prove uniform convergence of a series, one must show that the difference between the function values and the limit function values can be made arbitrarily small for all input values by choosing a large enough number of terms in the series. This can be done using mathematical techniques such as the Cauchy criterion or the Weierstrass M-test.

What are some applications of uniform convergence of series?

Uniform convergence of series has many applications in mathematics and physics. For example, it is used in Fourier analysis to approximate functions with trigonometric series, in numerical analysis to approximate solutions to differential equations, and in probability theory to prove the convergence of random variables.

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