EXERCISES 1. What is a power series?
11.8 EXERCISES 1. What is a power series?
n苷 兺 n!
1 n苷 兺 1 2 ⴢ 4 ⴢ 6 ⴢ ⴢ 共2n兲
(a) What is the radius of convergence of a power series?
How do you find it?
(b) What is the interval of convergence of a power series? 2n
25. 兺 2 26. x n苷 兺 n苷 1 n 2 n 共ln n兲 How do you find it? 2
3–28 Find the radius of convergence and interval of convergence
n苷 兺 1
of the series.
3. 兺 x 4. 兺 x n !x n
n苷 1 sn
n苷 0 1 28. 兺
n苷 1
n苷 兺 1 n 3 6. 兺 sn x
5. n
29. 冘 n苷 0 c n 4 n苷 n 1 If is convergent, does it follow that the following
series are convergent?
n苷 兺
7. x 8. n n n
n苷 兺 1 (a)
n苷 兺 c
(b)
0 n苷 兺 0
10 n x 9. n 兺 n
1 n n苷 2 n 10. n苷 兺 1 n 3 30. Suppose that 冘
n苷 0 c n x converges when 4 and diverges when x苷 6 . What can be said about the convergence or diver-
gence of the following series?
s n n n苷 兺 1 5 n n (a) c n (b) c n 8
n苷 1 5
n苷 兺 0 n苷 兺 0
x 13. 2n
n苷 兺
n苷 兺 0 共2n兲!
2 4 n ln n
(c) 兺 c n
n 14. n n
(d) 兺 c n 9
n苷 0 n苷 0
31. If is a positive integer, find the radius of convergence of k
15. n
n苷 兺 0 n 2 1 n苷 兺 0 the series
共n!兲 k
nn
n苷 兺 1 sn
n苷 兺 1 4 n
17. 18. n苷 0 共kn兲!
32. Let and be real numbers with p q
. Find a power series
n苷 兺 1 n 3 n
whose interval of convergence is
n苷 兺 1 b n ,
n苷 兺 3 1 Is it possible to find a power series whose interval of n 1 convergence is ? Explain.
CHAPTER 11 INFINITE SEQUENCES AND SERIES
34. Graph the first several partial sums s n 共x兲 of the series 冘 n苷 0 x n , ; CAS
(c) If your CAS has built-in Airy functions, graph on the A
same screen as the partial sums in part (b) and observe mon screen. On what interval do these partial sums appear to
together with the sum function f , on a com-
how the partial sums approximate . A
be converging to f 共x兲 ?
37. A function is defined by f
35. The function J 1 defined by
f 2 2x 3 4
n苷 兺 0 n !
that is, its coefficients are c 2n 苷 1 and c J 苷 1 共x兲 苷 2 for all n 0 .
Find the interval of convergence of the series and find an is called the Bessel function of order 1.
explicit formula for f 共x兲 .
38. If , where f 共x兲 苷 冘 n苷 0 c n x n c 4 苷 c n for all n 0 , find the ;
(a) Find its domain.
(b) Graph the first several partial sums on a common interval of convergence of the series and a formula for f 共x兲 . screen.
39. Show that if lim
ⱍ 苷 c n ⱍ c , where c苷 0 , then the radius
CAS
(c) If your CAS has built-in Bessel functions, graph J 1 on the
of convergence of the power series 冘 c n x n is R苷 1 兾c .
same screen as the partial sums in part (b) and observe
how the partial sums approximate . n J
40. Suppose that the power series
satisfies c n 苷 0
c n 兾c 1 exists, then it is equal The function defined by A
for all . Show that if 36. n lim
to the radius of convergence of the power series.
has radius of convergence 2 and the 2 3
x 3 x 6 x 9 41. Suppose the series 冘 c x n
series 冘 d n x n has radius of convergence 3. What is the radius
of convergence of the series 冘 共c n n 兲x n ?
is called the Airy function after the English mathematician and astronomer Sir George Airy (1801–1892).
42. Suppose that the radius of convergence of the power series
冘 (a) Find the domain of the Airy function. n c n x is . What is the radius of convergence of the power R
; 2n (b) Graph the first several partial sums on a common screen. series ? 冘 c n x