a) Find the Maclaurin series for the function f(x)=11+x b) Use differentiation of power series and the result of part a) to find the Maclaurin series for the function g(x)=1(x+1)2 c) Use differentiation of power series and the result of part b) to find the Maclaurin series for the function h(x)=1(x+1)3 d) Find the sum of the series ∑n=3∞n(n−1)2n This is a Taylor series problem, I understand parts a - c but I do not understand how to do part d where the answer is 72

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a) Find the Maclaurin series for the function  f(x)=11+x  b) Use differentiation of power series and the result of part a) to find the Maclaurin series for the function  g(x)=1(x+1)2  c) Use differentiation of power series and the result of part b) to find the Maclaurin series for the function  h(x)=1(x+1)3  d) Find the sum of the series  ∑n=3∞n(n−1)2n  This is a Taylor series problem, I understand parts a - c but I do not understand how to do part d where the answer is 72

Nathalie Redfern · Accepted Answer

It took me a while to get this, but finally I got it. Had to try a lot of tricks to get this series.  We have the series for 1(x+1)3 as shown below:  f(x)=1(1+x)3=∑n=0∞12(−1)n(n+1)(n+2)xn  We can similarly write a series for 1(1−x)3 as shown below:  f(x)=1(1−x)3=∑n=0∞12(n+1)(n+2)xn  Now, since the index  of the summation in our question starts from n=3, we could shift the index of this series to n=3. I tried that, but it didnt

Jeffrey Jordon · Accepted Answer

Answer is given below (on video)

a) Find the Maclaurin series for the function f(x)=frac11+x b) Use differentiation of power series and the result of part a) to find the Maclaurin ser

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