(Solution Library) a. Show that ∑_n=1^∞ (n(n+1))/(x^n)=(2 x^2)/((x-1)^3) for |x|>1 by differentiating the identity ∑_n=1^∞ x^n+1=fracx^21-x
Question: a. Show that
\[\sum_{n=1}^{\infty} \frac{n(n+1)}{x^{n}}=\frac{2 x^{2}}{(x-1)^{3}}\]for \(|x|>1\) by differentiating the identity
\[\sum_{n=1}^{\infty} x^{n+1}=\frac{x^{2}}{1-x}\]twice, multiplying the result by \(x\), and then replacing \(x\) by \(1 / x\)
b. Use part (a) to find the real solution greater than 1 of the equation
\[x=\sum_{n=1}^{\infty} \frac{n(n+1)}{x^{n}}\]
Deliverable: Word Document 
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