MATH 152, Fall 2014
Week In Review
Week 9
ln(2 + en )
.
3n
(−3)n
2. Determine if the following sequence is convergent
.
n!
3
n
3. Determine if the following sequence is convergent
.
n!
1 · 3 · 5 · · · · · (2n − 1)
4. Determine if the following sequence is convergent
.
(2n)n
1. Determine if the following sequence is convergent
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
1
. Assuming that an is convergent,nd its limit
an−2
is convergent. If so, the nd its
f or
n ≥ 3. Let an =
fn+1
and
fn
1
f or n ≥ 1, satises 0 < an ≤ 2 and is decreasing. Deduce
7. Show that the sequence a1 = 2 an+1 =
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 .
9. 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
10. 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
11. Write out the partial fraction decomposition of the function 4
. Do not determine the numerical
x + x3 + 2x2
values of the coecients.
12. Evaluate the integral
Z
18 − 2x − 4x2
dx .
x3 + 4x2 + x − 6
13. Evaluate the integral
Z
x4
dx .
x4 − 1
14. Evaluate the integral
Z
x3 − 2x2 + x + 1
dx .
x4 + 5x2 + 4
15. Evaluate the integral
Z
16. Evaluate the integral
Z
17. Evaluate the integral
Z
4
x2
0
2
−2
0
1
dx .
+x−6
1
dx .
x2 − 1
∞
√
1
dx .
x(1 + x)
18. If the innite curve y = e−x ,
x ≥ 0, is rotated about the x − axis, nd the area of the resulting surface.
19. 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
20. 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.
21. Find
Z
22. Find
Z
23. Find
Z
24. Find
Z
25. Find
Z
1
(5 − 4x − x2 )5/2
et
x2
−∞
Z
Z
0
dx.
dx.
1
x(lnx)2
dx.
9
1
(x − 9)1/3
2
1
4x − 5
1
27. Find
1
16x2 − 9
xe2x
e
26. Find
√
1
∞
dt.
p
9 − e2t
dx.
dx.
dx.