IIT JEE , AIEEE Forum - Mathematics , Differential Calculus

Ronaldooo

Lim (1/

n)(_[r=1]

^[r=4n] 1/

r) =?????
n->inf

Please help me out!!!

computer001

devide outside by rt(n) and multiply inside by rt(n)
now u can do gnl method n convert it 2 integration

Ronaldooo

What is gnl method and how to convert limit into integration??Can you plzzzz solve it fully??I will be grateful to you if u do so.

Ronaldooo

computer001,plzz explain!!

Ronaldooo

Anyone????????

sandeepramesh

Note: Computer001 has already said this so dont rate :) Rate him :D

multiply n divide by sqrt(n) it becomes

lim nto infinity sum 1 to 4n 1/sqrt(r/n) * 1/n

= integral 0 to 4 1/sqrt(x) dx = 2 sqrt(x) |0 to 4 = 4 :)

For further referrences, see limits by integration method

Ronaldooo

But when you multiply and divide by root(n),the exp. would become root(n/r)*(1/n) right?

sandeepramesh

thats what i said :) i/n becomes dx and 1/sqrt(r/n) becomes 1/sqrt(x)

layman

$lim_{n\rightarrow \infty} \frac{1}{\sqrt{n}} \sum^{r=4n}_{r=1} \frac{1}{\sqrt{r}}$

= $lim_{n\rightarrow \infty} \frac{1}{n} \sum^{r=4n}_{r=1} \sqrt{\frac{n}{r}}$

= $\int_{0}^{4} \frac{1}{\sqrt{x}}\,dx$

=4

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