11.8 Power Series

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11.8 Power Series

Power Series

A power series is a series of the form

where x is a variable and the c_n’s are constants called the coefficients of the series.

For each fixed x, the series (1) is a series of constants that we can test for convergence or divergence.

A power series may converge for some values of x and diverge for other values of x.

Power Series

The sum of the series is a function

f(x) = c₀ + c₁x + c₂x² + ^{. . .} + c_nxⁿ + ^{. . .} whose domain is the set of all x for which the series

converges. Notice that f resembles a polynomial. The only difference is that f has infinitely many terms.

For instance, if we take c_n = 1 for all n, the power series becomes the geometric series

xⁿ = 1 + x + x² + ^{. . .} + xⁿ + ^{. . .}

which converges when –1 < x < 1 and diverges when

| x | ≥ 1.

Power Series

In fact if we put in the geometric series (2) we get the convergent series

but if we put x = 2 in (2) we get the divergent series

Power Series

More generally, a series of the form

is called a power series in (x – a) or a power series centered at a or a power series about a.

Notice that in writing out the term corresponding to n = 0 in Equations 1 and 3 we have adopted the convention that (x – a)⁰ = 1 even when x = a.

Notice also that when x = a, all of the terms are 0 for n ≥ 1 and so the power series (3) always converges when x = a.

Example 1

For what values of x is the series n!xⁿ convergent?

Solution:

We use the Ratio Test. If we let a_n, as usual, denote the nth term of the series, then a_n = n!xⁿ. If x ≠ 0, we have

=

By the Ratio Test, the series diverges when x ≠ 0. Thus the given series converges only when x = 0.

Power Series

We will see that the main use of a power series is that it provides a way to represent some of the most important

functions that arise in mathematics, physics, and chemistry.

In particular, the sum of the power series,

, is called a Bessel function.

We know that the sum of a series is equal to the limit of the sequence of partial sums. So when we define the Bessel function as the sum of a series we mean that, for every real number x, where

Power Series

The first few partial sums are

Power Series

Figure 1 shows the graphs of these partial sums, which are polynomials. They are all approximations to the function J₀, but notice that the approximations become better when

more terms are included.

Partial sums of the Bessel function J₀

Figure 1

Power Series

Figure 2 shows a more complete graph of the Bessel function.

For the power series that we have looked at so far, the set of values of x for which the series is convergent has always turned out to be an interval.

Figure 2

Power Series

The following theorem says that this is true in general.

The number R in case (iii) is called the radius of

convergence of the power series. By convention, the radius of convergence is R = 0 in case (i) and R = in case (ii).

Power Series

The interval of convergence of a power series is the

interval that consists of all values of x for which the series converges.

In case (i) the interval consists of just a single point a.

In case (ii) the interval is ( , ).

In case (iii) note that the inequality | x – a | < R can be rewritten as a – R < x < a + R.

Power Series

When x is an endpoint of the interval, that is, x = a ± R,

anything can happen—the series might converge at one or both endpoints or it might diverge at both endpoints.

Thus in case (iii) there are four possibilities for the interval of convergence:

(a – R, a + R) (a – R, a + R] [a – R, a + R) [a – R, a + R]

The situation is illustrated in Figure 3.

Figure 3

Power Series

We summarize here the radius and interval of convergence for each of the examples already considered in this section.