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thanks of viewing.
Well, it's convenient for looking at convergence and limits, no?
yes. it is.
so what do you think of my question?.
Well, we can sum up the series (if the limit of partial sums exists)
we can? how do we know if the limit of partial sums exists?
so your saying if a series Converges it partial sums exists
Well, if a series diverges, there wouldn't be any convergence, right?
And you wouldn't be able to add it up however hard you tried
yes BUT because its a infinite series whether it converges or diverges we cant sum it all up?
That's what I'm saying, if it converges then you can (usually) add it up (analytically).
i see what your saying sure. i just don't see the idea of converges or diverges has anything to do with the fact that a series Converges and there for the partial sums exists
If the series converges then you can take a limit
All well and good that you are saying the facts. :) i really want to understand why we can take a limit when series converges and not diverges.
am sorry am being a pain.
the limit of a divergent series is infinite and some other limits might not exist (there are shades of convergence, absolute, conditional...)
You must have a way, analyytically, to approach a limit (even if you do not actually ever reach it)
I am assuming you have studied calculus, the limit concept there is similar.
i see . if we say that x-----> 2 x approaches 2 . what comment would you have?
i have studied CAL but in a very mechanical way!
You can say it and it implies that the limit is 2 but doesn't say anything about how you got that.
do we care how we get there . is that important for us?
How do we know that your limit is valid?
The limit of 1 + 1/2 +1/4 + 1/8 ......+1/2^n has a limit of 2 Do you believe me?
well you told me thats
Ok, it's not 2 it's 3
no its not !
Anyway, the point is that it converges and it is fairly easy to establish that 2 is an upper bound and with a little more work that the sum is actually 2
And now I am off to bed, I'm afraid.....
visually . |dw:1348445954284:dw|
thanks you Rock. :)