POWER SERIES
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38.60
Show that In 2 = 1
By Problem 38.38, In(l + x) =
for |*| <1. By the alternating series theorem, the series
converges when x — 1. By Abel's theorem, In 2 =
Abel's theorem reads: Let f(x) =
for - r < x < r. If the series converges for
then lim /(*) exists and is equal to
38.61
Show that
By Problem 38.36,
for |j»;| < I. The series converges for x = l, by the alternating series test. Hence, by Abel's theorem (Problem 38.60),
38.62
Show that the converse of Abel's theorem fails; that is, show that if
for HO, if the series
has radius of convergence r, and if
f(x) = b, then
does not necessarily converge.
Consider
with radius of convergence 1.
but
is
not convergent.
38.63 Find a power series for sin
2 x.
By Problem 38.59,
Hence,
Adding 1 eliminates the constant term — 1, yielding
So
38.64 Find a power series for
is the sum of the geometric series with first term
and ratio
for
38.65 Find a power series for
and
for
38.66 Find power series solutions of the differential equation xy" + y' - y = 0.
Let
Then
Further,
and
Now,
Therefore, a l = a 0 and (n + 1)
2
«,, + 1 = «„. Hence,
and, in general,
is an arbitrary constant.
Thus,
where a 0
38.67 Find the interval of convergence of
Use the ratio test.
x. The function defined by this series is denoted /„(*) and is called a Bessel function of the first kind of order
zero.
Hence, the series converges for all
l + 2*-3jc
2 = (l-*)(l + 3.x),