Question

$\boxed{\sf \lim_{n \to \infty} (\frac{1}{1-n^2}+\frac{2}{1-n^2} +...\: \frac{n}{1-n^2} }$

- Need a step-by-step answer!
- Thank you!​

Answer 1

Answer:

Let X,Y and Z be three jointly continuous random variables with joint PDF

fXYZ(x,y,z)={

1

3

(x+2y+3z) 0≤x,y,z≤1 0 otherwise

Find the joint PDF of X and Y, fXY(x,y).

Answer 2

$\begin{gathered}\\ \sf\Rrightarrow \lim_{n\to \infty}\left(\dfrac{1}{1-n^2}+\dfrac{2}{2-n^2}\dots \dfrac{n}{1-n^2}\right)\end{gathered}$

Take LCM as 1-n^2

$\begin{gathered}\\ \sf\Rrightarrow \lim_{n\to \infty}\left(\dfrac{1+2+3\dots n}{1-n^2}\right)\end{gathered}$

1+2..n=n(n+1)/2

$\begin{gathered}\\ \sf\Rrightarrow \lim_{n\to \infty}\left(\dfrac{\dfrac{n(n+1)}{2}}{1-n^2}\right)\end{gathered}$

$\begin{gathered}\\ \sf\Rrightarrow \lim_{n\to \infty}\dfrac{n(n+1)}{2(1-n^2)}\end{gathered}$

$\begin{gathered}\\ \sf\Rrightarrow \lim_{n\to infty}\dfrac{n(1+n)}{2(1-n)(1+n)}\end{gathered}$

$\begin{gathered}\\ \sf\Rrightarrow \lim_{n\to \infty}\dfrac{n}{2(1-n)}\end{gathered}⇛n→∞lim2(1−n)n</p><p>\begin{gathered}\\ \sf\Rrightarrow \dfrac{\infty}{2-\infty}\end{gathered}$

$\begin{gathered}\\ \sf\Rrightarrow \dfrac{-1}{2}\end{gathered}$