2 The radius of convergence

The most important statement one can make about a power series is that there exists a number, R , called the radius of convergence, such that if   x < R  the power series is absolutely convergent and if   x > R  the power series is divergent. At the two points x = R and x = R the power series may be convergent or divergent.

Key Point 11

Convergence of Power Series

For a power series p = 0 b p x p with radius of convergence R then

the series converges absolutely if x < R

the series diverges if x > R

the series may be convergent or divergent at x = ± R

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For any particular power series p = 0 b p x p the value of R can be obtained using the ratio test. We know, from the ratio test that p = 0 b p x p is absolutely convergent if

lim p b p + 1 x p + 1 b p x p = lim p b p + 1 b p x < 1 implying | x | < lim p b p b p + 1 and so R = lim p b p b p + 1 .

Example 2
  1. Find the radius of convergence of the series

    1 + x 2 + x 2 3 + x 3 4 +

  2. Investigate what happens at the end-points x = 1 , x = + 1 of the region of

      absolute convergence.

Solution
  1. Here 1 + x 2 + x 2 3 + x 3 4 + = p = 0 x p p + 1

    so

    b p = 1 p + 1 b p + 1 = 1 p + 2

    In this case,

    R = lim p p + 2 p + 1 = 1

    so the given series is absolutely convergent if x < 1 and is divergent if x > 1 .

  2. At x = + 1 the series is 1 + 1 2 + 1 3 + which is divergent (the harmonic series). However, at x = 1 the series is 1 1 2 + 1 3 1 4 + which is convergent (the alternating harmonic series).

    Finally, therefore, the series

    1 + x 2 + x 2 3 + x 3 4 +

    is convergent if 1 x < 1.

Task!

Find the range of values of x for which the following power series converges:

1 + x 3 + x 2 3 2 + x 3 3 3 +

First find the coefficient of x p :

b p = 1 3 p

Now find R , the radius of convergence:

R = lim p b p b p + 1 = lim p 3 p + 1 3 p = lim p ( 3 ) = 3.

When x = ± 3 the series is clearly divergent. Hence the series is convergent only if 3 < x < 3.