8.6. POWER SERIES
527
for at least −1 < x < 1.
It is a straightforward exercise to check that the power series
∞
k=0
(−1)
k x 2k+1
2k + 1
converges both when x = −1 and when x = 1; in each case, we have an alternating series
with terms
1
2k+1 that decrease to 0, and thus the interval of convergence for the series
expansion for arctan(x) in Equation (8.27) is −1 ≤ x ≤ 1.
Activity 8.31.
Find a power series expansion for ln(1 + x) centered at x = 0 and determine its interval
of convergence. (Hint: Use the Taylor series expansion for
1
1 + x
centered at x = 0.)
⊳
Summary
In this section, we encountered the following important ideas:
• A power series is a series of the form
∞
k=0
a k x
k .
• We can often assume a solution to a given problem can be written as a power series,
then use the information in the problem to determine the coefficients in the power series.
This method allows us to approximate solutions to certain problems using partial sums
of the power series; that is, we can find approximate solutions that are polynomials.
• The connection between power series and Taylor series is that they are essentially the
same thing: on its interval of convergence a power series is the Taylor series of its sum.