Q-9E - Fundamentals Of Differential Equations And Boundary Value Problems

Question

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

 

Step-by-Step Solution

Answer

The required expansion is.

Step 1: 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

 

Step 2: conclusion

The sum therefore is given by