Sorry for necroing an old thread, but I've gone back, and am reviewing my infinite series knowledge. I thought the previous discussions here might be useful for my query.
My question is, can we generalize an infinite series so that it can be denoted by:
\Sigma ^{\infty} _{n=m}a_n where m is any...
What if the limit of S(n) as n approaches infinity is known to be S, I've read that the function S(n-1) as n approaches infinity is also S. How do we verify this? My idea is that, after finding an N such that: |S(n)-S|<ε whenever n>N; and we adjust ε so that we find a 1<N.
So that...
I think defining the sum of an infinite series using the associated sequence of partial sums is pretty ingenious. The limit of a sequence is already defined, which is the limit of the sequence function, I think this makes everything pretty neat.
You can think of the sequence of partial sums as...
Ah, yes, yes, now that I have completely rewritten the theorem, and with your clarification I now understand better.
I sometimes feel like I'm hunting Easter eggs when I'm looking at theorems.
Sorry, I might not remember it very well, I'll just do a direct quote from a textbook then.
Woah, I now realize I'm completely wrong, my 'theorem' up there was a product of my cluttered head, so please excuse it. Here goes the correct theorem:
If \Sigma _{n=1} ^{\infty} a_n and \Sigma _{n=1}...
For the series such that: \Sigma _{n=1} ^{\infty} a_n =\Sigma _{n=1} ^{\infty} b_n A certain theorem says that these series are equal even if a_n = b_n only for n>m. That is, even if two infinite series differ for a finite number of terms, it will still converge for the same sum. I am thinking...
Hm, well I could think that obviously if n≥N then this implies n+1≥N (if N>0 and n is restricted to positive integers). Does that mean I can say: if for any ε>0, and N>0 such that |S(n)-S|<ε whenever n≥N is true by hypothesis, then |S(n+1)-S|<ε whenever n+1≥N is also true?
The catch is I've...
I have a question about limits at infinity, particularly, about a limit I have seen in the context of infinite series convergence.
Let's say we have an infinite series where the the sequence of partial sums is given by {S(n)} and also, it is convergent and the sum is equal to S. Then we know...