Question

What is the coefficient of x^n in the expansion of :

Answer 1

\[(1+x+2x^2+3x^3+...+nx^n)^2\]

Answer 2

Please have you tried to use the induction principle?

Answer 3

this is (1-x)^(-2) so expand (Binomial) (1-x)^(-4).

Answer 4

I got this formula:
\[1 + 2{x^2} + 3{x^3} + ... + n{x^n} = n{S_n} - \sum\limits_{n = 1}^{n - 1} {{S_k}} ,\quad with:\;{S_n} = 1 + x + {x^2} + ... + {x^n}\]

Answer 5

\[\begin{align}(1+x+2x^2+3x^3+...+nx^n)^2 &= \left(1+\sum\limits_{k=1}^{n} kx^{k}\right)^2 \\~\\
&= \left( 1+\sum\limits_{k=1}^{n} x(x^{k})'\right)^2\\~\\
&= \left( 1+x\left[\sum\limits_{k=1}^{n} x^{k}\right]'\right)^2\\~\\
& = \left(1+ x^2\left[\dfrac{x^n-1}{x-1}\right]'\right)^2\\~\\
&=
\cdots\end{align}\]

Answer 6

Thank you everybody ! :)