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Math 152, Spring 2008-copyright Joe Kahlig
1. (5 points) Find an equation of the sphere with center at (1, 4, −6) and passes through the point (−1, 3, 1)
2. (5 points) Let a =< 1, 0, −2 > and b =< −1, −2, −4 >. Compute the angle between a and b.
3. (5 points) Suppose that the nth partial sum of the series
∞
P
an is given by sn =
n=2
∞
P
5n + ln(n)
. Will the series
n
an will converge or diverge? If the series converges, find the sum of the series if possible.
n=2
4. (14 points) Find the radius and the interval of convergence for
∞
X
n=2
nxn
+ 1)
4n (n2
5. (5 points) For the Taylor series for g(x) = ln(x + 4) centered at a = −2, find the coefficient of the (x + 2)3 term?
6. (8 points) Use the series to approximate this integral so that the error < 10−2 . Use the fewest number of terms
possible to attain the prescribed accuracy.
Z0.5
3
e−x dx =
0
7. (16 points) Find the power series representation for these functions centered at a = 0. Also give the radius of
convergence.
sin(2x)
x2
4
(b) g(x) = 2
9x + 4
(a) f (x) =
8. (14 points) Determine if these series converge or diverge. If it converges, find its sum. Fully justify your answer.
(a)
n ∞ n
X
−1
1
−4
7
5
9
n=0
(b)
∞
X
1
(n
+
1)(n
+ 2)
n=5
9. (14 points) Determine if the series is divergent, absolutely convergent, or just convergent. Fully justify your
answer.
(a)
∞
X
(−1)n
√
n
n=1
(b)
∞
X
(2n)!
n!(n + 1)!
n=1
10. (14 points) Determine if these series converge or diverge. Fully justify your answer.
∞
X
ln(n)
(a)
n3
n=1
(b)
∞
X
(−1)n n
√
n2 + 5
n=1
Check the back of the page for more problems.