Question

I need a pedagogical exercise of the application of the Weierstrass \(M\)-test. Anyone?

Answer 1

Here is the theorem's statement:
(Weierstrass \(M\)-test) Suppose that \((f_n)\) is a sequence of functions defined on \(S\) and \(M_n\) is a sequence of nonnegative numbers such that\[|f_n(x)|\leq M_n\text{, for all }x\in S\text{ and all }n\in\mathbb{N}.\]If \(\sum M_n\) converges, then \(\sum f_n\) converges uniformly on \(S\).

Answer 2

Sorry this is out of my league.

Answer 3

Same

Answer 4

pedagogical exercise
Does that mean you are trying to teach this theorem and need a specific example?

Answer 5

That's correct!

Answer 6

why was i tagged in this? im retarded

Answer 7

What type of class is this for? Just curious

Answer 8

This is real analysis.

Answer 9

Yuck Real Analysis, my nightmares are returning

Answer 10

loool

Answer 11

seems like firstly, if fn is <= to Mn, then it seem intuitive that if Mn converges, then fn could too. So you want an application of this idea? Like an example problem so that you could teach someone else? Like create a word problem?

Answer 12

http://en.wikipedia.org/wiki/Uniform_convergence
Can't you use something like the examples shown?

Answer 13

But the only example Wikipedia gives for a uniformly convergent sequence of functions is on \(\mathbb{C}\). :(

Answer 14

http://yourmathsolver.blogspot.com/
this looks like a cool site to check out

Answer 15

Suppose\[f _{k}(x)=1/k^2(\sin(x)+\cos(x))\]So, now we have\[|f _{k}(x)|\le2/k^2=M _{k}\]Further, note\[\sum_{K=1}^{\infty}M_k<\infty\]So \[\sum_{K=1}^{\infty}|f _{k}(x)|<\infty\]

Answer 16

You think that one is OK?

Answer 17

Give me a second to work it out.

Answer 18

That one works. :) Thanks!

Answer 19

No sweat. The typing is the hardest part.....LOL