I just need to get a better understanding on what a telescoping series is.
I think I can recognize it easily enough, but.. please take a look at this:
"Determine if the following series diverges or converges. If the series converges and it is possible, tell me what the sum of the series is. Identify what kind of series it is or the test you used to validate your answer."
I don't know how to make all the fancy math symbols, but it's
(sigma from n=1 to infinity)(5/n - 5/n+1)
Plugging in the first few values,
(5/1 - 5/2) + (5/2 - 5/3) + (5/3 - 5/4) ....
So everything from n+1 to infinity cancels each other out (-5/2 + 5/2, -5/3 + 5/3, etc, etc). We are left with just 5/1, or 5.
So.. the sum of the series is just 5 + 0 + 0 + 0 ... = 5.. right?
I would say it converges and the sum is 5. Is this right?
My notes say the sum is the first number (5/1) minus the limit ... I don't really understand.
Any help is appreciated.
When are considering a series we cannot think of that as a ‘infinite string of numbers’.
We look at the sequence of partial sums: . (partial sums are finite strings)
We look at the limit .
So series convergence is really about sequence convergence, the sequence of partial sums.