420 STEP 5. Build Your Test-Taking Confidence
- The correct answer is (D).
Rewrite
∑∞
n= 1
3 n+^2
4 n
as
∑∞
n= 0
(3)^2
(
3 n
4 n
)
= 9
∑∞
n= 0
(
3
4
)n
= 9
(
1 +
3
4
+
(
3
4
) 2
+
(
3
4
) 3
+···
)
. Note
that 1+
3
4
+
(
3
4
) 2
+
(
3
4
) 3
is a geometric
series with a ratio of
3
4
. Thus the infinite series
is
a 1
1 −r
=
1
1 −
3
4
=4 and 9
∑∞
n= 0
(
3
4
)n
= 36.
- The correct answer is (D).
The limh→ 0
ln(x+h−3)−ln(x−3)
h
is the
definition of the derivative for the function
y=ln(x−3); therefore, the limit is equal to
y′=
1
x− 3
.
- The correct answer is (C).
The Taylor polynomial of third degree about
x=aisf(a)+f′(a)(x−a)+
f′′(a)
2!
(x−a)^2 +
f′′′(a)
3!
(x−a)^3. In this case, the third-degree
polynomial aboutx=2is− 1 +(1)(x−2)−
2
2!
(x−2)^2 +
6
3!
(x−2)^3 or− 3 +x−(x−2)^2 +
(x−2)^3.
- The correct answer is (A).
Note thatf′′>0 on the intervals (−∞,−4)
and (0, 4). Thus, the graph of the functionf is
concave up on these intervals. Similarly,
f′′<0 and concave down on the intervals
(−4, 0) and (4,∞). Therefore,fhas a point of
inflection atx=−4, 0, and 4. - The correct answer is (C).
Rewrite
√
k
kn
as
k
(^12)
kn
or
1
kn−
12 , which is ap-series.
In order for the series to converge,
(
n−
1
2
)
must be greater than 1. Thus,n−
1
2
> 1
orn>
3
2
.
- The correct answer is (C).
Sincef′(x)>0,f(x) is increasing, and since
f′′>0,f(x) is concave up. The graph of
f(x) may look like the one below. The right
Riemann sum contains the largest rectangles.
y f
0 12 x
Right Riemann Sum
Right
Trapezoidal
Midpoint
Left
34