Question

question

Answer 1

find the sum of the series.


\(\large \color{black}{\begin{align}

1+\dfrac{4}{7}+\dfrac{9}{7^2}+\dfrac{16}{7^3}+\cdots \ \cdots+\infty^{\normalsize +} \hspace{.33em}\\~\\


\end{align}}\)

Answer 2

\(\large \color{black}{\begin{align}


\sum_{n=1}^{\infty} \dfrac{n^2}{7^{n-1}}\hspace{.33em}\\~\\


\end{align}}\)

Answer 3

is this valid


\(\large \color{black}{\begin{align}

\sum_{n=1}^{\infty} \dfrac{n^2}{7^{n-1}}\hspace{.33em}\\~\\

\implies \dfrac{\sum_{n=1}^{\infty} n^2}{\sum_{n=1}^{\infty} 7^{n-1}}\hspace{.33em}\\~\\
\end{align}}\)

Answer 4

Recall that for \(|x|<1\),
\[f(x)=\frac{1}{1-x}=\sum_{n=0}^\infty x^n\]
Take the first derivative:
\[f'(x)=\frac{1}{(1-x)^2}=\sum_{n=1}^\infty nx^{n-1}\]
Multiply by \(x\):
\[xf'(x)=\frac{x}{(1-x)^2}=\sum_{n=1}^\infty nx^{n}\]
Take the derivative again:
\[f'(x)+xf''(x)=\frac{1+x}{(1-x)^3}=\sum_{n=1}^\infty n^2x^{n-1}\]
Set \(x=\dfrac{1}{7}\) and you get your sum.