518 Chapter 8 • Infinite Series of Real Numbers
- Complete Part 2 of the proof of Theorem 8.6.14.
- Find the Taylor series for ln x (x > 0) about 1, and find its interval
of convergence. [Write ~ = l-(Lx) and apply the method of Example
8.6.18.J - Use known power series to find Maclaurin series representing each of the
following functions, and find the interval of convergence in each case:
1
(a) 1 + x3
x2
(b) 1 + x3
x2
(e) (1 + x3)2(c) x^2 ln(l + x)
x4
(f) (1 + x3)31- Use the Maclaurin series for --and its derivatives to find each of the
l-x
following sums. Determine the interval of convergence where appropriate.
= = = k = k^2
(a) k"f1 kxk (b) k"f1 k2xk (c) k"f1 3k (d) k"f1 3k - Explain why lx l, lnx, and ..Ji have no power series representations about
0.
12. Show that xP has no power series representation about 0 if p is a real
number other than a positive integer.13. Prove that a Maclaurin series representing an even function has only even-
powered terms, and a Maclaurin series representing an odd function has
only odd-powered terms.14. Prove Corollary 8.6.20. [Apply Theorem 8.6.19 to f(u), where u = px.]- Prove the assertions made in Example 8.6.21 (b).
- Find a power series representation of f 0 "' tan-^1 t dt and determine its
interval of convergence (Abel's theorem may be helpful). Use this result
to show that
(Partial fractions may also be helpful.)- For each of the following series, determine the values of x for which the
series converges, and the values of x for which it converges absolutely.
= (-l)kk2
(a) L (x - 3)k
k=O
(b) f co~kx
k=O