I'm sorry if you have a diffuclty reading this. I am not sure how i would go about displaying this in HTML
above the Sigma there is an (n) and below (i=1) on the right side of the sigma this is displayed (1+ 2i/n) (2/n). Let s(n) = the above problem
Find the limit of s(n) as n -> infinity
How do I solve this problem? Could someone help me out step by step please?
VonNemo19's expression is the result of factoring; the is an upper limit, not an index, so it can come out. Now you should know both of these identities:
The second identity can be proved by induction, or, better yet, by the ingenius method of Gauss, letting be the sum:
s = 1 + 2 + 3 + ... + n
s = n + ... + 3 + 2 + 1
Each line has n terms, and the terms of each line, added together, equal n+1. Hence 2s = n(n+1), or s= n(n+1)/2.
Using these identities, you should be able to solve the problem easily, letting n approach infinity.
You can see that will be a factor of every term, so therefore we can pull it out in front of the sigma symbol
Now, understand that if I were to sum the remaining terms together, they would expand together. Heres what I mean...
So, you can see how I could rearange the terms to be like this...
So, it's like I have two sums
One is , and the other is .
So, then this gives
Now notice the second term inside the brackets. I can factor out
Now from the formulas
. and we get
Taking the limit