a)Expand f(x) = (x+x^2)/(1-x)^3 as a power series.
b)Use part a) to find the sum of the series: sum of (n^2)/(2^n) with n = 1 to infinity.
Can someone help me with these? Thanks a lot.
Use a Maclaurin series (a Taylor series expanded about 0):
$\displaystyle f(x) = \frac{x + x^2}{(1 - x)^3}$
$\displaystyle f^{\prime}(x) = \frac{x^2 + 4x + 1}{(1 - x)^4}$
$\displaystyle f^{\prime \prime}(x) = -\frac{2(x^2 + 7x + 4)}{(1 - x)^5}$
$\displaystyle f^{\prime \prime \prime}(x) = \frac{6(x^2 + 10x + 9)}{(1 - x)^6}$
etc.
So
$\displaystyle f(x) \approx f(0) + \frac{1}{1!} f^{\prime}(0) \cdot x + \frac{1}{2!} f^{\prime \prime}(0) \cdot x^2 + \frac{1}{3!} f^{\prime}(0) \cdot x^3 + ~ ...$
$\displaystyle f(x) \approx x + 4x^2 + 9x^3 + ~ ...$
Can you spot the pattern? Can you see how it relates to the sum you are trying to find?
-Dan