MATH 152, Fall 2013
Week In Review
Week 11
1. Determine wether the following series are absolutely convergent.
(a)
∞
X
(−3)n
.
n3
n=1
(b)
∞
X
(−3)n
.
n!
n=1
∞
X
(−1)n+1
.
2n + 1
n=1
√
∞
X
(−1)n−1 n
(d)
.
n+1
n=1
(c)
(e)
∞
X
(−1)n+1 5n+1
.
(n + 1)2 4n+2
n=1
(f)
∞
X
cos(nπ/6)
√
.
n n
n=1
2. For which of the following series is the ratio test inconclusive ?
∞
X
1
(a)
.
3
n
n=1
∞
X
(−3)n−1
√
.
n
n=1
√
∞
X
n
(c)
.
1
+
n2
n=1
(b)
3. For which positive integers k is the series
∞
X
(n!)2
(kn)!
n=1
convergent ?
4. (a) Show that
∞
X
xn
is convergent for all x.
n!
n=0
(b) Deduce that limn→∞
5. Suppose that the series
xn
=0
n!
∞
X
n=0
cn xn is convergent when x = 4 and diverges when x = 6. Is the series
convergent ? What can be said about the convergence or divergence of the following series ?
(a)
(b)
(c)
∞
X
cn 8n .
n=0
∞
X
n=0
∞
X
cn (−3)n .
(−1)n cn 9n .
n=0
∞
X
n=0
cn
6. If
∞
X
cn 4n is convergent , does it follow that the following series are convergent?
n=0
∞
X
(a)
(b)
(c)
cn (−2)n .
n=0
∞
X
n=0
∞
X
cn (3)n .
cn (−4)n .
n=0
7. Find the radius and interval of convergence of the seriws .
(a)
∞
X
xn
.
n2
n=1
(b)
∞
X
(−1)n xn
.
n2n
n=1
(c)
∞
X
n
(2x − 1)n .
n
4
n=0
∞
X
(d)
(e)
(−1)n
n=1
∞
X
(x − 1)n
√
.
n
(x − 4)n
.
n5n
n=1
(f)
∞
X
2n (x − 3)n
.
n+3
n=0
(g)
∞
X
(x + 1)n
.
n(n + 1)
n=1
∞
X
(h)
(i)
n!(2x − 1)n .
n=1
∞
X
nxn
.
1 · 3 · 5 · · · · · (2n − 1)
n=1
8. If k is a positive integer, nd the radius of convergence of the series
∞
X
(n!)k n
x
(nk)!
n=0
.
9. A function f is dened by:
f (x) = 1 + 2x + x2 + 2x3 + x4 + 2x5 + x6 + · · ·
That is, its coecients are c2n = 1 and c2n+1 = 2 for all n ≥ 0. Find the interval of convergence of the series
and nd an explicit formula for f (x) .
10. If f (x) =
f (x).
∞
X
n=0
cn xn where cn+4 = cn for all n ≥ 0, nd the interval of convergence of the series and a formula for
11. Find a power series representation for the following functions, and determine the interval of convergence.
1
.
1 + 4x2
1
(b) f (x) = 4
.
x + 16
1 + x2
.
(c) f (x) =
1 − x2
1
(d) f (x) =
.
(1 + x)3
(a) f (x) =
(e) f (x) = xln(1 + x) .
(f) f (x) =
x2
.
(1 − 2x)2
12. Evaluate the indenite integral as a power series
Z
x
dx .
1 + x5
Z
tan−1 (x)
(b)
dx .
x
Z
(c) tan−1 (x2 )dx .
(a)
13. Use a power series to approximate the denite integral to six decimal places.
(a)
Z
1/2
tan−1 (x2 )dx .
0
(b)
Z
(c)
Z
0.5
0
1/3
1
dx .
1 + x6
x2 tan−1 (x4 )dx .
0
14. Dened the function f by the power series
f (x) =
∞
X
xn
.
n!
n=0
(a) Show that f is solution of the dierential equation
f 00 (x) + f (x) = 0 f or
.
(b) Prove that f (x) = ex .
all
x