Question

Help please!!

simplify the following:

`\displaystyle \lim_(n\to \infty ) \frac{1 + 2 + 3 + \dots {\dots} +n }{ {n}^(2) }`

`\text{ note: must provide explanation}`
​

Answer 1

`(1)/(2)`

Explanation:

we are given a limit

and we want to simplify it

notice that, the numerator is in a sequence of sum of natural number

recall that,

`\rm \displaystyle \: 1 + 2 + 3 + \dots {\dots }+ n = (n(n + 1))/(2)`

so substitute:

`\displaystyle \lim_(n\to \infty ) \frac{ (n(n + 1))/(2) }{ {n}^(2) }`

now recall L'Hôpital's rule

`\displaystyle \lim_(x \to c) (f(x))/(g(x)) = \lim_(x \to c) (f'(x))/(g'(x))`

first simplify the complex fraction:

`\displaystyle \lim_(n\to \infty ) (n+ 1)/(2n)`

apply L'Hôpital's rule:

`\displaystyle \lim_( n\to \infty ) ( (d)/(dn) n+ 1)/( (d)/(dn) 2n)`

simplify:

`(1)/(2)`

and we are done:

Answer 2

Explanation:

solution given:

The given expression takes a form
`( \infty )/( \infty )`

when n=
`\infty`

so,

`\displaystyle \lim_(n\to \infty ) \frac{1 + 2 + 3 + \dots {\dots} +n }{ {n}^(2) }`

`\displaystyle \lim_(n\to \infty )\frac{n(n+1)}{ 2{n}^(2) }`

`\displaystyle \lim_(n\to \infty )(n+1)/( 2n)`

`(1)/(2)`

`\displaystyle \lim_(n\to \infty )[1+(1)/( n)]`

=
`(1)/(2)`(1+
`(1)/( \infty ))`

=
`(1)/(2)` is your answer