Practice problems for Exam 3
Calculus II, MA 112
Instructions: This is a no-maple exam. You may use your calculator for simple calculations.
(6 pts) 1. Determine the limit if the limit exists. If the limit does not exist, write DNE.
µ
µ ¶n
¶n
2n3 + n + 1
1
2
lim 1 −
lim
lim
3
n→∞
n→∞
n→∞
3n + 1
n
5
lim (−1)n
n→∞
n+1
n
lim
n→∞
(10 pts) 2. Show that the telescoping series
∞
X
n=2
cos(πn)
n
2
lim √
n→∞
n
1
is convergent and equal to 1 by
n(n − 1)
(a) finding a simplified expression for the partial sum, sn (note that you will need to apply
1
the PFD to
- show your work)
n(n − 1)
Write your answer here: sn =
(b) and then determine lim sn =
n→∞
∞ µ ¶n
X
1
3
(10 pts) 3. Show that the geometric series
is convergent and equal to by
3
2
n=0
(a) finding a simplified expression for the partial sum sn (note that you need to use the
factorization 1 − xn+1 = (1 − x)(1 + x + x2 + · · · + xn ) which we discussed in class)
Write your answer here: sn =
(b) and then find lim sn .
n→∞
(20 pts) 4. Find the following by hand. Show all steps.
Z
(a) x arctan(x) dx
(b)
Z
5x + 1
dx
+x−2
x2
(20 pts) 5. Do the following by hand. Show all steps.
(a) lim
x→0
x + e2x − 1
x + sin(x)
(b) Determine whether the following improper integral is convergent or divergent. If convergent, find its value.
Z ∞
x
dx
2
0 x +1
(32 pts) 4. To the left of the series, write C if the series is convergent or D if the series is divergent.
To the right of the series, give your reason - you need not do the details - simply put the
method (or theorem) which you would use to justify your answer (for example, the series
is geometric, the integral test, comparison test, comparison test, terms do not converge to
zero...). However, you need to give the information required in the test: for example, the
geometric series with r = 12 , the comparison test with an = ... and bn = ..., the p series test
with ...
∞ µ ¶n
X
3
a)
7
n=0
b)
∞
X
3
n2
n=1
∞
X
1
√
c)
n
n=1
e)
∞
X
n=1
f)
∞
X
1
(1 − )n
n
e−n
n=1
g)
∞
X
n=0
h)
∞
X
n=1
1
(n + 1)(n + 2)
n
2n − 1
(10 pts) 5. Suppose that an > 0. If s4 is the 4th partial sum of
∞
X
an , show how to think of s4
n=1
geometrically. That is, plot a1 , a2 , a3 , a4 , . . . and then show how s4 can be thought of
geometrically. (We used this idea when discussing the integral test and also the comparison
test).
(10 pts) 6. A ball is dropped from a height of 6 feet and begins bouncing. The height of each bounce
is 34 the height of the previous bounce. Find the total vertical distance traveled by the ball.