Q-10E - Fundamentals Of Differential Equations And Boundary Value Problems

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.

Step-by-Step Solution

Answer

The required expansion is,

Step 1: 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

Step 2: 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,