MATH 152, Spring 2014
Week In Review
Week 10
1. Determine if the following sequence is convergent
2. Determine if the following sequence is convergent
3. Determine if the following sequence is convergent
4. Determine if the following sequence is convergent
ln(2 + en )
.
3n
(−3)n
.
n!
3
n
.
n!
1 · 3 · 5 · · · · · (2n − 1)
.
(2n)n
1
5. Determine if the following sequence a1 = 1 an+1 =
1 + an
limit.
6. The Fibonacci sequence is dened by f1 = 1,
show an−1 = 1 +
f2 = 1,
f or
n≥1
fn = fn−1 + fn−2
is convergent. If so, the nd its
f or
n ≥ 3. Let an =
1
. Assuming that an is convergent,nd its limit.
an−2
fn+1
and
fn
1
7. Show that the sequence a1 = 2 an+1 =
f or n ≥ 1, satises 0 < an ≤ 2 and is decreasing. Deduce
3 − an
that the sequence is convergent, and nd its limits.
1
8. A sequence is dened recursively by a1 = 1 an+1 = 1 +
f or n ≥ 1. Find the rst eight terms of the
1 + an
sequence. What do you notice about the odd terms and even terms ? By cinsidering the odd and even terms
separately, show that an is convergent and nd its limit .
∞ X
1
1
√ −√
9. Determine the sum of the series
.
n
n+1
n=1
10. Let an =
∞
X
2n
. Determine whether {an } is convergent. Determine whether
an is convergent.
3n + 1
n=1
8 16
32
+
+
+ · · · is convergent or divergent. If it is convergent, nd its sum.
5 25 125
∞ X
1
12. Determine whether the series
is convergent or divergent. If it is convergent, nd its sum.
e2n
n=1
11. Determine whether the series 4 +
13. 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
14. Determine whether the series
15. Determine whether the series
16. Determine whether the series
17. 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
√
n
is convergent or divergent. If it is convergent, nd its sum.
1 + n2
18. Determine whether the series
∞
X
1
is convergent or divergent. If it is convergent, nd its sum.
1
+
2−n
n=1
19. 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
20. Find the values for x such that the series
21. Find the values for x such that the series
∞
X
1
is convergent. Find the sum for those values of x.
n
x
n=1
∞
X
tann x is convergent. Find the sum for those values of x.
n=1
∞
X
22. Find the sum of the series
1
.
(4n
−
1)(4n
− 3)
n=1
23. Find the sum of the series
∞
X
n2 + 3n + 1
.
(n2 + n)2
n=1
24. If the n − th partial sum of a series
∞
X
an is
n=1
sn =
nd an and
∞
X
n−1
n+1
an .
n=1
25. 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
26. What is the value for c if
∞
X
(1 + c)−n such that the series is convergent ?
n=2
27. Suppose that
∞
X
an
n=1
∞
X
1
an 6= 0, is known to be convergent. Prove that
is divergent.
a
n=1 n
28. Write out the partial fraction decomposition of the function
values of the coecients.
x4 + x2 + 1
. Do not determine the numerical
(x2 + 1)(x2 + 4)2
29. Write out the partial fraction decomposition of the function
numerical values of the coecients.
19x
. Do not determine the
(x − 1)3 (4x2 + 5x + 3)2
x3 + x2 + 1
30. Write out the partial fraction decomposition of the function 4
. Do not determine the numerical
x + x3 + 2x2
values of the coecients.
31. Evaluate the integral
Z
18 − 2x − 4x2
dx .
x3 + 4x2 + x − 6
32. Evaluate the integral
Z
x4
dx .
x4 − 1
33. Evaluate the integral
Z
x3 − 2x2 + x + 1
dx .
x4 + 5x2 + 4
34. Evaluate the integral
Z
35. Evaluate the integral
Z
36. Evaluate the integral
Z
4
0
1
dx .
x2 + x − 6
2
−2
0
1
dx .
−1
x2
∞
√
1
dx .
x(1 + x)
37. If the innite curve y = e−x ,
x ≥ 0, is rotated about the x − axis, nd the area of the resulting surface.
38. Find the area of the surface obtained by rotating the curve x = acos3 (θ), y = asin3 (θ),
x − axis.
0 ≤ θ ≤ π/2 about the
39. Set up the integral giving the surface area of the ellipsoid obtained by rotating the ellipse x = acos(θ), y =
bsin(θ), a ≥ b, about the y − axis.