I'm really stumped on this question and am looking for help. If anybody could help me figure this out and show me some working I would owe you indefinitely, plus i'll give you a virtual cookie :3
The picture of the question is here: http://tinypic.com/r/4gmc84/6
Any help is greatly appreciated!
The picture of the question is here: http://tinypic.com/r/4gmc84/6
Any help is greatly appreciated!
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lim(n→∞) Σ(i = 1 to n) (5/n^3) (i - 1)^2
= lim(n→∞) (5/n^3) * Σ(i = 1 to n) (i^2 - 2i + 1)
= lim(n→∞) (5/n^3) * [Σ(i = 1 to n) i^2 - 2 * Σ(i = 1 to n) i + Σ(i = 1 to n) 1]
= lim(n→∞) (5/n^3) * [n(n+1)(2n+1)/6 - 2 * n(n+1)/2 + n], via summation formulas
= lim(n→∞) 5 * [(n+1)(2n+1)/(6n^2) - (n+1)/n^2 + 1/n^2]
= lim(n→∞) 5 * [(1/6)(1 + 1/n)(2 + 1/n) - 1/n]
= 5 * [(1/6) 1 * 2 - 0]
= 5/3.
I hope this helps!
= lim(n→∞) (5/n^3) * Σ(i = 1 to n) (i^2 - 2i + 1)
= lim(n→∞) (5/n^3) * [Σ(i = 1 to n) i^2 - 2 * Σ(i = 1 to n) i + Σ(i = 1 to n) 1]
= lim(n→∞) (5/n^3) * [n(n+1)(2n+1)/6 - 2 * n(n+1)/2 + n], via summation formulas
= lim(n→∞) 5 * [(n+1)(2n+1)/(6n^2) - (n+1)/n^2 + 1/n^2]
= lim(n→∞) 5 * [(1/6)(1 + 1/n)(2 + 1/n) - 1/n]
= 5 * [(1/6) 1 * 2 - 0]
= 5/3.
I hope this helps!