Search results

1. Infinite Series question

Hm, does that mean, the sequence of partial sums of the said series will also have 'm' as its starting index?
2. Infinite Series question

I'd really appreciate it if someone would reply. Is my question too trivial?
3. Is this statement valid?

About 0.999...=1, is representing it as a geometric infinite series a valid proof?
4. Infinite Series question

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...
5. Limit on infinity

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...
6. Series, what are they for?

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...
7. Infinite Series question

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.
8. Infinite Series question

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}...
9. Differentiating f(-x)?

Chain rule: \frac{d[f(g(x))]}{dx} = \frac{d[f(g(x))]}{d[g(x)]} \frac{d[g(x)]}{dx} . In your case, g(x) = -x.
10. Infinite Series question

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...
11. Limit on infinity

I think that's same things as what I've said. Haha, sometimes I have a hard time wrapping my head around stuff like this. Thanks anyway.
12. Limit on infinity

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...
13. Limit on infinity

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...