POWER SERIES
337
38.87
Let f(x) =
Show that
(a 0 + a l x + • • • + a n x" + •••) = (1 + x + x
2 + • • • + x" + • • -)(a a + a,:H
a n x" + •••). The terms of
this product involving x" are O O AC" + a,x • x" ' H
1- a n _ 2 x" " • x
2 + a n _ l x" ' • x + a n x". Hence, the coefficient of AC" will be a 0 + a, + • • • + a,,.
38.88 Find a power series for
In (1 - x) = - (x + x
2 /2 + x
3 /3 + •••) by Problem 38.50. Hence, by Problem 38.87, the coefficient of x" in
Hence,
38.89
Write the first four terms of a power series for (sin *)/(! — x).
By Problem 38.87, the coefficient of x" in (sinx)/
(1 - x) will be the sum of the coefficients of the power series for sin x up through that of x . Hence, we get
(smx)/(l-x) = x + x
2 + i*
3 + I*
4 + ••-.
38.90
Find the first five terms of a power series for (tan 'je)/(l — x).
tan"
1 x = x-x*/3 +x
s /5 -x
1 /l+ ••• for |*|<1, by Problem 38.36. Hence, by Problem 38.87, we
obtain (tan"
1 jc)/(l - x) = x + x
2 + f jc
3 + |x
4 + gx
5 + •••.
38.91
Find the first five terms of a power series for (cos x)/(I - x).
cosx = l-x
2 /2\ + x
4 /4\-x
6 /6\ + x
s /8\
. Hence, by Problem 38.87, (cos*)/(l - x) = 1 + x + |jr +
ije'+fi*
4 + "-.
38.92
Use the result of Problem 38.87 to evaluate
We want to find
so that a 0 + a l + •• • + a = n + 1. A simple choice is a, = 1. Thus, from
we obtain
38.93 If /(*) =
a x", show that the even part of /(*), E(x) = \ [/(*)+ f(-x)], is
and the odd
part of/(jc), 0(x)=±[f(x)-f(-x)],
is
Hence,
and
38.94
Evaluate x
2 /2 + jt
4 /4 + • • • + x
2t /2k + • • •.
This is the even part (see Problem 38.93) of the series /(*) = -In (1 - x) = x + x
2 /2 + x
3 /3 + x
4 /4 +••-.
Hence the given series is equal to
38.95
Use Problem 38.93 to evaluate
This is the even part of e*, which is (e* + e ') 12 = cosh x. (This problem was solved in the reverse direction
in Problem 38.43.)
38.%
Find
by power series methods.
By Problem 38.58, sin x - x = -*
3
/3! + jc
s
/5! - x
7 /T. + •••. Hence,
(sin x-x) /x
3 = -1 /3! + x
2 /5! -
*
4
/7! + ---, and lim (sinx-x)/x
3 = -1/3! = -$.
x—»0