Proving Convergence of Series in a Given Norm

In summary, the process to determine the convergence or divergence of a series in a given norm is not mentioned in the book. However, one can use the method of determining whether a series is Cauchy to show its convergence. Depending on the specific series, there may be different methods for proving convergence or divergence. The example given is the series f_{n}(t) in the space (C[0,1],||\cdot||_{\infty}). Despite asking for help, the original poster did not receive any assistance and requested for the post to be moved to the Homework Help Forum if necessary.
  • #1
Somefantastik
230
0
Given a sequence, how does one prove that the associated series in convergent or not, in a given norm? For example,

[tex]\sum_{k=0}^{\infty}a_{k} [/tex] in [tex]||\cdot|| [/tex]

The process to do this is not in my book; I'm told how to determine whether a series is cauchy, but I'm not sure how to use that to show it's convergent.
 
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  • #2
depending on the series there will be a different method for proof of convergence or divergence. is there a specific series you are speaking of?
 
  • #3
I have several, in several spaces.

[tex] f_{n}(t) \ in \ (C[0,1],||\cdot||_{\infty}) [/tex]

is an example of one.
 
  • #4
got nothing for you man. sorry.
 
  • #5
I don't even understand your notation. good luck though
 
  • #6
bump, can someone please give this another look? I'd like to work these problems, but my book is not helpful and my campus is closed this week :(

If there is no help, can a moderator move this post into the Homework Help Forum?
 

1. How do you define convergence of a series in a given norm?

Convergence of a series in a given norm means that the sequence of partial sums of the series approaches a specific value in the given norm as the number of terms in the series increases.

2. What is the difference between absolute convergence and conditional convergence in a given norm?

Absolute convergence of a series in a given norm means that the series converges in the norm regardless of the order in which the terms are added. Conditional convergence means that the series converges in the norm only when the terms are added in a specific order.

3. How do you prove the convergence of a series in a given norm?

To prove the convergence of a series in a given norm, one can use the limit comparison test, ratio test, or root test. These tests compare the given series to a known convergent or divergent series, and if the comparison is valid, it can be concluded that the given series also converges or diverges in the same norm.

4. Can a series converge in one norm but diverge in another?

Yes, it is possible for a series to converge in one norm but diverge in another. This is because different norms may have different definitions of convergence and may measure the distance between terms differently.

5. Are there any special cases where a series will always converge in a given norm?

Yes, there are special cases where a series will always converge in a given norm. For example, if the terms of the series decrease in magnitude and approach 0, the series will converge in any norm. Another example is a geometric series with a common ratio less than 1, which will always converge in any norm.

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