Question

Question 10: In Problems, find the power series expansion for f(x)+g(x), given the expansions for f(x) and g(x).

10.

Short answer

The required expansion is,

Step-by-step solution

Power series expansion 

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

 changing of index

In order to obtain sum we need to have common index for both the series f(x) and g(x).

Changing the index of f(x). Let z = n - 1 = > n = z + 1

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

Changing the index of g(x),

If we substitute other variable in g(x), it will change the order of the variable (x-1), therefore,

The summation of the series f(x)+g(x) is therefore represented by

Hence, the required expansion is,