11.9

11.9 Representations of Functions

as Power Series

Representations of Functions as Power Series

We start with an equation that we have seen before:

We have obtained it by observing that the series is a geometric series with a = 1 and r = x.

But here our point of view is different. We now regard

Equation 1 as expressing the function f(x) = 1/(1 – x) as a sum of a power series.

Representations of Functions as Power Series

and some partial sums

Figure 1

Example 1

Express 1/(1 + x²) as the sum of a power series and find the interval of convergence.

Solution:

Replacing x by –x² in Equation 1, we have

Example 1 – Solution

Because this is a geometric series, it converges when

| –x²| < 1, that is, x² < 1, or | x | < 1.

Therefore the interval of convergence is (–1, 1). (Of course, we could have determined the radius of convergence by

applying the Ratio Test, but that much work is unnecessary here.)

cont’d

Differentiation and Integration of

Power Series

Differentiation and Integration of Power Series

The sum of a power series is a function whose domain is the interval of convergence of the series.

We would like to be able to differentiate and integrate such functions, and the following theorem says that we can do so by differentiating or integrating each individual term in the series, just as we would for a polynomial.

This is called term-by-term differentiation and integration.

Differentiation and Integration of Power Series

Example 4

We have seen the Bessel function

is defined for all x.

Thus, by Theorem 2, J₀ is differentiable for all x and its

derivative is found by term-by-term differentiation as follows: