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8.6. POWER SERIES
Exercises
1. We can use power series to approximate definite integrals to which known techniques of
integration do not apply. We will illustrate this in this exercise with the definite integral
1
0
sin(x 2 ) ds.
(a) Use the Taylor series for sin(x) to find the Taylor series for sin(x 2 ). What is
the interval of convergence for the Taylor series for sin(x 2 )? Explain.
(b) Integrate the Taylor series for sin(x 2 ) term by term to obtain a power series
expansion for
sin(x 2 ) dx.
(c) Use the result from part (b) to explain how to evaluate
1
0
sin(x 2 ) dx. Determine
the number of terms you will need to approximate
1
0
sin(x 2 ) dx to 3 decimal
places.
2. There is an important connection between power series and Taylor series. Suppose f
is defined by a power series centered at 0 so that
f (x) =
∞
k=0
a k x
k .
(a) Determine the first 4 derivatives of f evaluated at 0 in terms of the coefficients
a k .
(b) Show that f (n) (0) = n!a n for each positive integer n.
(c) Explain how the result of (b) tells us the following:
On its interval of convergence, a power series is the Taylor series of its sum.
3. In this exercise we will begin with a strange power series and then find its sum. The
Fibonacci sequence { f n } is a famous sequence whose first few terms are
f 0 = 0, f 1 = 1, f 2 = 1, f 3 = 2, f 4 = 3, f 5 = 5, f 6 = 8, f 7 = 13, · · · ,
where each term in the sequence after the first two is the sum of the preceding two
terms. That is, f 0 = 0, f 1 = 1 and for n ≥ 2 we have
f n = f n−1 + f n−2 .
Now consider the power series
F(x) =
∞
k=0
f k x
k .