## The Calculus Primer (2011)

### Part XI. Expansion of Functions

### Chapter 41. POWER SERIES

**11—11. Convergence of Power Series.** Any series of the form

*a*_{0} + *a*_{1}*x + a*_{2}*x*^{2} + *a*_{3}*x*^{3} + … + *a _{n}*

_{−1}

*x*

^{n}^{−1}+

*a*… ,

_{n}x^{n}+where *a*_{0}, *a*_{1}, *a*_{2}, … are constants, is known as a *power series* in *x.* Such series are important in higher mathematics, for they are often used in finding the values of a given function. In fact, we shall develop special methods for expressing any ordinary function in terms of a series.

Whether a power series converges or diverges depends upon the particular value assigned to *x.* Thus every power series converges when *x* = 0. A power series may converge for all values of *x,* or for no values of *x* other than zero. In general, however, a power series converges for some values of *x* besides *x* = 0, and diverges for other values of *x.*

To determine the values for which a power series converges, we use the following ratio test. Thus in a power series, if = *L*, then the series

(a)converges for all values of *x* such that |*x| < L*;

(b)diverges for all values of *x* such that |*x| > L*;

(c)no test if |*x| = L.*

**11—12. Interval of Convergence of a Power Series.** The values of *x* for which a power series converges are said to constitute the *interval of convergence* for the series. It can be proved that this interval, when plotted, will always have zero as the center. A series may or may not converge for the value of *x* at either end point of its interval of convergence. For all other values of *x,* the series is divergent.

EXAMPLE 1. Find the values of *x* for which the series is convergent:

*Solution.* Here *a*_{0} = 1, *a*_{1} = , *a*_{2} = , etc.

Hence the series converges for *|x|* < 1, that is, for –1 < *x* < 1; it diverges for *|x| >* 1, that is, for –1 > *x >* 1. To test the end points:

when *x* = 1, we have

which is divergent;

when *x* = − 1, we have

which is also divergent.

Hence the end points are not included in the interval of convergence.

EXAMPLE 2. Find the values of *x* for which the series is convergent:

Thus the series converges for *|x|* < 1, or for − 1 < *x* < 1.

To test the end points:

when *x* = 1, we have

which is convergent;

when *x* = − 1, we have

which is divergent.

Hence for the original power series, the end point *x =* + 1 is included in the interval of convergence, but the other end point, − 1, is not included.

EXAMPLE 3.Find the interval of convergence of

*Solution.* Here *a*_{0} = 1, *a*_{1} = 1, *a*_{2} = , etc.

Therefore the series converges for *|x|* < ∞, that is, for — ∞ < *x* < ∞, or for all positive and negative values of *x.*

**EXERCISE 11—4**

*Determine the values of x for which the following series are convergent:*