Question

Question : find the power series expansion_for given the expansions for f(x) and g(x).

Short answer

The required expansion is.

Step-by-step solution

Find the sum of functions

We know that the power series representation of the sum of two power series, given byis simply obtained by term-wise addition:

For the given series

In order to obtain sum we need to change the index of function g(x)

Let z=n-1=n=z+1

Since index is just a dummy variable, we can replace z with n; the series g(x) thus becomes

conclusion

The sum therefore is given by