Question

Evaluate the given limit by first recognizing the indicatd sum as a Riemann sum associated with a regular partiion of [0,1] and then evaluating the corresponding integral.

(a) lim (1+2+3+......+n)/n^2 as n approaches to positive infinity.

Answer 1

we can evaluate this limit easily without doing that

\[\sum_{k=1}^nk=\frac{n(n+1)}{2}\] so
\[\frac{1}{n^2}\sum_{k=1}^nk=[frac{1}{n^2}\frac{n(n+1)}{2}=\frac{n^2+n}{2n^2}=\frac{1}{2}+\frac{1}{n}\]
take limit as n goes to infinity and get
\[\frac{1}{2}\]

Answer 2

stupid typo
\[\frac{1}{n^2}\sum_{k=1}^nk=\frac{1}{n^2}\frac{n(n+1)}{2}=\frac{n^2+n}{2n^2}=\frac{1}{2}+\frac{1}{n}\]

Answer 3

or you could recognize this as
\[\int_0^1 xdx=\frac{1}{2}\]

Answer 4

how can you tell it is x?