I have encountered many difficulities with the following question, I think I have made if more complicated than it seems. If anyone has any insight on how I would go about it, parts a & b, it would be greatly appreciated.
Starting with the geometric series \(\displaystyle 1\, +\, x\, +\, x^2\, +\, x^3\, +\, ...\, (-1\, <\, x\, 1),\) find the power series for \(\displaystyle \dfrac{x}{(1\, -\, x)^2}\) in powers of x.
a) Where is the series valid?
b) Using the result in (a), find the sum \(\displaystyle \displaystyle{\sum_{n = 1}^{\infty}\, \frac{n}{2^n}}\).
Starting with the geometric series \(\displaystyle 1\, +\, x\, +\, x^2\, +\, x^3\, +\, ...\, (-1\, <\, x\, 1),\) find the power series for \(\displaystyle \dfrac{x}{(1\, -\, x)^2}\) in powers of x.
a) Where is the series valid?
b) Using the result in (a), find the sum \(\displaystyle \displaystyle{\sum_{n = 1}^{\infty}\, \frac{n}{2^n}}\).
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