POWER SERIES
331
38.46
Approximate
dt correctly to three decimal places
By Problem 38.45,
This is an alternating series. Hence, we must find the
least n such that
n 2:4. So we may use
38.47
For a fixed integer k > 1, evaluate
In Problem 38.37, we found that, for
Hence,
for
Then
and the desired value is
38.48
Evaluate
In Problem 38.47, let k = 2. Then
38.49
Use power series to solve the differential equation y' - -xy under the boundary condition that y = \ when
je = 0.
Let
Differentiate:
Comparing coefficients, we get
fl,=0,
and
So
Since y = 1 when x = 0, we know that a n = 1. Now. a, = a, = a, = • • • = 0.
Also, 2a 2 -— a 0 = -l,
Similarly,
Then, 4a 4 = — « 2 ,
Further, 6a 6 = —a 4 , a 6 =
8a 8 = -a 6 ,
and, in general,
So
By Problem 38.41,
38.50
Find a power series representation for In (1 — x).
Substitute —x for x in the series of Problem 38.38: ln(l — jc) =
for
38.51
Find a power series representation for In
Bv Problems 38.38 and 38.50. for
In
In
In
38.52
Use the power series for In
to approximate In 2.
By Problem 38.51, In
When
So In
Using the first three terms, we get
(The correct value to four
decimal places is 0.6931.)
38.53
Use the power series for In
to approximate In 3.
When
As in Problem 38.52, In3 = 2
(The correct value to four places is 1.0986.)