MATH 152, Fall 2014
Week In Review
Week 11
∞ X
1
1
√ −√
.
1. Determine the sum of the series
n
n+1
n=1
2. Let an =
∞
X
2n
. Determine whether {an } is convergent. Determine whether
an is convergent.
3n + 1
n=1
32
8 16
+
+
+ · · · is convergent or divergent. If it is convergent, nd its sum.
5 25 125
∞ X
1
4. Determine whether the series
is convergent or divergent. If it is convergent, nd its sum.
e2n
n=1
3. Determine whether the series 4 +
5. Determine whether the series
∞
X
[2(0.1)n + (0.2)n ] is convergent or divergent. If it is convergent, nd its sum.
n=1
6. Determine whether the series
7. Determine whether the series
8. Determine whether the series
9. Determine whether the series
∞
X
n2
is convergent or divergent. If it is convergent, nd its sum.
3(n + 1)(n + 2)
n=1
∞
X
1
is convergent or divergent. If it is convergent, nd its sum.
n(n
+ 2)
n=1
∞
X
1
is convergent or divergent. If it is convergent, nd its sum.
2−1
4n
n=1
∞
X
√
n=1
10. Determine whether the series
n
is convergent or divergent. If it is convergent, nd its sum.
1 + n2
∞
X
1
is convergent or divergent. If it is convergent, nd its sum.
1 + 2−n
n=1
11. Find the values for x such that the series
∞
X
(x − 3)n is convergent. Find the sum for those values of x.
n=1
∞
X
1
12. Find the values for x such that the series
is convergent. Find the sum for those values of x.
n
x
n=1
13. Find the values for x such that the series
∞
X
tann x is convergent. Find the sum for those values of x.
n=1
∞
X
14. Find the sum of the series
1
.
(4n + 1)(4n − 3)
n=1
15. Find the sum of the series
∞
X
n2 + 3n + 1
.
(n2 + n)2
n=1
16. If the n − th partial sum of a series
∞
X
an is
n=1
sn =
nd an and
∞
X
n=1
an .
n−1
n+1
17. If the n − th partial sum of a series
∞
X
an is
n=1
sn = 3 − n2−n
nd an and
∞
X
an .
n=1
18. What is the value for c if
∞
X
(1 + c)−n such that the series is convergent.?
n=2
19. Suppose that
∞
X
an
an 6= 0, is known to be convergent. Prove that
n=1
∞
X
1
is divergent.
a
n=1 n
20. Determine whether 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
21. For which of the following series is the ratio test inconclusive ?
(a)
∞
X
1
.
3
n
n=1
∞
X
(−3)n−1
√
.
n
n=1
√
∞
X
n
(c)
.
1
+
n2
n=1
(b)
22. For which positive integers k is the series
∞
X
(n!)2
(kn)!
n=1
convergent ?
23. (a) Show that
∞
X
xn
is convergent for all x.
n!
n=0
(b) Deduce that limn→∞
24. Suppose that the series
xn
=0
n!
∞
X
n=0
cn x
n
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 ?
∞
X
n=0
cn
(a)
(b)
(c)
∞
X
cn 8n .
n=0
∞
X
cn (−3)n .
n=0
∞
X
(−1)n cn 9n .
n=0
25. If
∞
X
cn 4n is convergent , does it follow that the following series are convergent?
n=0
(a)
(b)
(c)
∞
X
cn (−2)n .
n=0
∞
X
n=0
∞
X
cn (3)n .
cn (−4)n .
n=0
26. Find the radius and interval of convergence of the series .
(a)
∞
X
xn
.
n2
n=1
(b)
∞
X
(−1)n xn
.
n2n
n=1
(c)
∞
X
n
(2x − 1)n .
n
4
n=0
(d)
(e)
∞
X
(−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
(h)
(i)
∞
X
n!(2x − 1)n .
n=1
∞
X
nxn
.
1 · 3 · 5 · · · · · (2n − 1)
n=1
27. If k is a positive integer, nd the radius of convergence of the series
.
∞
X
(n!)k n
x
(nk)!
n=0
28. 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) .
29. If f (x) =
f (x).
∞
X
cn xn where cn+4 = cn for all n ≥ 0, nd the interval of convergence of the series and a formula for
n=0
30. 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
(a) f (x) =