POWER SERIES
335
38.75 For what values of x can sin x be replaced by x if the allowable error is 0.0005?
By Problem 38.58, sin x = x —
Since this is an alternating
series, the error is less than the magnitude of the first term omitted. If we only use*, the error is less than |.x:|
3
/3!.
So we need to have
38.76
Use power series to evaluate
38.77 Approximate
(sin x)lx dx correctly to six decimal places.
By Problem 38.58,
So
Note that 9 • 9! = 3,265,920, and, thereHence, it suffices to calculate
fore, the next term,
which
yields 0.946083.
38.78
Use the multiplication of power series to verify that e* e ' = 1.
Refer to Problems 38.39 and 38.40.
But the binomial theorem gives, for k s 1, 0 = (1 - 1)* =
Hence
the coefficient of x in (1) is zero, for all k > 1; and we are left with
38.79 Find the first five terms of the power series for e* cos x by multiplication of power series.
Hence,
and
38.80
Find the first five terms of the power series for e* sin x.
38.81
Find the first four terms of the power series for sec x.
Let
Then
Now we equate coefficients. From the constant coefficient, 1 = a 0 . From the coefficient of
From the coefficient of
hence,
From the coefficient of
From the coefficient of
From the coefficient of
hence,
Thus,
From the coefficient of
hence
38.82
Find the first four terms of the power series for e*/cos x by long division.
Write the long division as follows: