336
CHAPTER 38
Hence, the power series for e'lcos x begins with 1 + x + x
2 + 2x
3 /3.
38.83 Find the first three terms of the power series for tan x by long division.
Arrange the division of sin x by cos x as follows:
Hence, the power series for tan* begins with
38.84 Evaluate the power series x + 2x
2 + 3x
3 + • • • + nx" + • • •.
By Problem 38.37, 1/(1 - x)
2 = I + 2x + 3x
2 + • • -. Therefore, x/(l - x)
2 = x + 2x
2 + 3x
3 + • • -.
38.85 Evaluate the power series
Let
Hence,
Now,
by Problem 38.84.
and
38.86 Write the first three terms of the power series for In sec x.
(lnsec*) = tanjc = *+ $x
3 + £x
5 + • •-, by Problem 38.83. Therefore, msec* = C + x~/2 + x
t l\2 +
When * = 0, In sec AT = 0. Hence, C = 0. Thus, In sec x = x
z /2 + x
4 /12 + *
6
/45 + • • •.