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CHAPTER 38
38.36
Show that
for
Integrate the power series of Problem 38.35 term by term, and note that tan ' 0 = 0.
38.37
Find a power series representation for
Method 1. By Problem 37.113, for
Differentiate this series
term by term. Then, for
Method 2.
38.38
Find a power series representation for ln(l + ;t) for |*|<1.
By Problem 38.34, for
Integrate term by term:
In
38.39
Show that
for all x.
Let
By Problem 38.3, f(x) is defined for all x. Differentiate term
by term:
Moreover, /(0) = 1. Hence, by Problem 24.72, f(x) = e*.
38.40
Find a power series representation for e
v .
By Problem 38.39,
Substitute -x for x: e *
38.41
Find a power series representation for e * '".
By Problem 38.40,
Substitute
for x. Then
38.42
Approximate \le correctly to two decimal places.
By Problem 38.40, e~" = 1 - x + x
2 /2l - *
3 /3! + • • •. Let x = l. Then lie = 1 - 1 + 1/2! - 1/3! + • • -.
Let us use the alternating series theorem here. We must find the least n for which l/n!<0.005= 555,
2006. So we can use 1 - 1 + k - 5 + a - lio = TIB = 0.3666••-. Sol/e = 0.37, correct to two
decimal places.
38.43
Find a power series representation for cosh x.
Using the series found in Problems 38.39 and 38.40, we have cosh
38.44
Find a power series representation for sinh x.
Since D T (cosh x) = sinh x, we can differentiate the power series of Problem 38.43 to get sinh x =
38.45
Find a power series representation for the normal distribution function
By Problem 38.41,
Integrate: