Properties of power series

 Properties of power series involve the following operations:

1) The addition and subtraction of power series

2) The multiplication of a power series by a constant or a power of the variable

3) The multiplication of two power series

4) The differentiation and integration of power series.

In the last post we represent certain functions in term of power series. The properties of power series help to facilitate this process. They allow us to find the power series representations of certain elementary functions by re-writing them in terms of functions of known series representations.

Combining power series

If we have two power series in the same interval of convergence, we can add or subtract these two power series to obtain a new series in the same interval of convergence. Similarly, we can multiply a power series by by a power of x that leads to a new power series. These properties allow us as we mentioned earlier to find the power series representation of certain functions knowing the power series representations of other functions . For example knowing the power series representation of f(x) = 1/1-x we can find the power representation of f(x) = 1/(1-x)² and y = 1/(x-1)(x-3).

Theorem

Example

Solution

for all x in the interval (-1,1).

According to the properties of combining power series above, the series

Practice