I. Determine whether the given series is absolutely convergent, conditionally convergent, or divergent. ∞ X (−1)n (2n + 5) √ 1. 3 n7 + 1 n=2
2. 3.
∞ X n=1 ∞ X n=1
1 √ √n ne (−1)n n(ln n)2
II. Determine the interval of convergence and radius of convergence of the following series:
1. 2.
∞ X (−1)n (x − 2)n n=0 ∞ X n=1
3.
(2n)! 3n (x + 2)n √ n+1
4.
∞ X (−1)n (x − 3)n n=1 ∞ X n=0
n ln n (x + 1)n 2n (n2 + 1)
III. Find the derivative and the integral of each of the following power series:
1.
∞ X
x
2n
2.
∞ X n=1
n=0
(−1)n
(x − 1)n n
IV. Find a power series representation of f at a if 1. f (x) =
x , 2+x
2. f (x) = tan−1 x, 3. f (x) = cos x, V. Approximate
√ 3
a=0 a=1
(Hint: Use the Taylor series formula.)
a = π/2
(Hint: Use the Taylor series formula.)
1.01 using the 3rd degree Taylor polynomial at 1 of
√ 3
x.