534 Chapter 8 • Infinite Series of Real Numbers
- Prove the following slightly stronger version of Theorem 8.7.5: If f is
infinitely differentiable on an open interval I containing c and 3 M > 0 3
oo J(n)( )
Vx EI, Vn EN, IJ(nl(x)I ~Mn, then Vx E J, f(x) = 'L::-- 1 c-(x-c)n.
n.
n=O - Use the method of Example 8. 7. 7 to find the Taylor series of cos x about c.
- Complete the proof of Case 2 of Theorem 8.7.10.
- Find each of the following integrals using power series. Can you find these
integrals without using power series?
(b) j e:x dx
- The Maclaurin series for ex, sin x, and cos x converge to these functions for
all real numbers. Assume that the same is true for all "complex" numbers
(real numbers in combination with the "imaginary" number i = yCT).
Derive the identity eix = cos x + i sin x and from it deduce Euler's famous
identity, ei'lr = -1. (If you have never seen this amazing identity before,
you may need a little time to let it sink in.) - Consider the function^13 J(x) = { e-~/x
2
if x "I-O }· If you have not
0 1f x = 0
worked out Exercise 6.6.16, do so now. Use the result to show that the
Maclaurin series of this function converges everywhere, but does not con-
verge to f(x) for any nonzero x.00 (2k )- Consider the function^14 f(x) = L cos k! x. Assuming that the succes-
k=l
sive derivatives off can be found by term-by-term differentiation,^15 show
that V odd k EN, j(k)(O) = 0 and that V even k EN, IJ(k)(O)i = e^2 k -1.
Show that this yields an infinitely differentiable function whose Maclaurin
series diverges everywhere except at 0.
1 - Show that the function^16 f(x) = -- 2 is infinitely differentiable every-
1 + x
where, but that its Maclaurin series converges only for lxl < l. - Prove that the sum by columns of the matrix given in Example 8.7.15 is
2, but the sum by rows diverges. - This example appears in a lmost every textbook on this subject.
- This example may be found in [61], page 256.
- This will be shown in Chapter 9.
- This function is suggested in [16], page 179.