338
CHAPTER 38
38.97 Evaluate x!2\ + x
2 /3l + x
3 /4\ + *
4 /5! + • • -.
Let f(x) = x/2l + x
2 /3l + x
3 /4\ + x
4 /5l + ---.
Then
xf(x) = x
2 /2\ + x
3 /3\ + x*/4\ + x
5 /5\ + ••• =
e'-x-l. Hence, /(*).= (e' - x - l)/x.
38.98 Assume that the coefficients of a power series
repeat every k terms, that is, a n+k = a n for all n.
Show that its sum is
Let
Then,
The series converges for |*|<1.
g(x) = a 0 + a lX + --- + a k ^x
k \
a n x" = g(x) + g(x)x
k + g(x)x
2k + • • • = g(x)(\ + x
k +
x
2k + •••) =
38.99 Evaluate
Let
Then
In
by
Problem 38.38.
Hence,
/(*) = f In (1 + x) dx = (1 + x) [In (1 + x) - 1] + C = (1 + x) In (1 + x) - x + C,.
When x = Q, f(x) = 0, and, therefore, (^=0. Thus, /(*) = (1 + *)ln (1 + x) - x.
38.100 Evaluate x*!4 + *
8
/8 + x
>2 /12 + x
l6 /l6+ • • -.
By Problem 38.50,
-In (1 - u) = u + u
2 /2 + u
3 /3 + «
4 /4+ • • -.
Hence,
-ln(l - x') = x
4 + x
s /2 +
x
>2 /3 +x
l6 /4+•••. Thus, the given series is -? m(l - *
4 ).
38.101 Evaluate
If f(x) is the function of Problem 38.85, then
38.102 Evaluate
38.103 For the binomial series (Problem 38.31),
•••, which is convergent for |*|<1, show that (1 + x)f'(x) = mf(x).
the coefficient of x" will be
In
Hence, (1 + x)f'(x) = mf(x).
38.104 Prove that the binomial series f(x) of Problem 38.103 is equal to (1 + x)
m .
Let
Then,
by Problem 38.103. Hence,
g(x) is a constant C. But f(x) = \ when * = 0, and, therefore, C = l. Therefore, f(x) = (\ + x}
m .
38.105 Show that
Substitute -x for x and \ for "i in the binomial series of Problems 38.103 and 38.104.
38.106 Derive the series
Substitute -x for x and - \ for m in the binomial series of Problems 38.103 and 38.104. (Alternatively, take
the derivative of the series in Problem 38.105.)
38.107 Obtain the series