Mathematics 55
2nd Semester 2010-2011
I.
1. Show that the sequence
n2 − ln n n3
converges by finding its limit.
2. What can be said about the convergence or divergence of the series
+∞ 2 X n − ln n ? n3 n=1
II. Find the sum of the series. 1.
+∞ n+1 X 2 n−1 3 n=1
2.
+∞ X
tan−1 n − tan−1 (n + 1)
n=1
III. Determine whether the given series converges or diverges. 1.
+∞ X sin n n2 + 1 n=1
4.
+∞ √ 3 X n−1 n=1
+∞ X
2.
(−1)n n2n en2 n=1
3.
+∞ X ln n n2 n=1
5.
n+1
+∞ X (−1)n−1 2n+1 (2n)! n=1
IV. Do as indicated. 1. Determine the radius and interval of convergence of the power series: (a)
+∞ X (−1)n xn−1 √ n+1 n=1
(b)
+∞ n X 3 (x + 1)n n n=1
2. Obtain a power series representation for: (a) f (x) =
x
(b) g(x) = ln |1 − 3x|
2
(2 + x2 )
3. Use the Maclaurin series for g(x) = cos x to evaluate the sum of the series 4. Give the third degree Taylor polynomial of f (x) =
√ 3
+∞ X (−1)n 9n . (2n)! n=0
x about -1 and use it to approximate
√ 3
−0.5.
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