Calculus I (Math161)
Exam 3 (Spring, 2016)
Department: Id number: Name:
1. (12 pts.) Let a1, . . . , a100 be real numbers such that
a1+· · · + a100= 0.
Then prove that lim ℓ→∞(a1 √ ℓ + 1 + a2 √ ℓ + 2 +· · · + a100 √ ℓ + 100) = 0.
2. (12 pts.) Find the values of p for which the series is convergent. ∞ ∑ n=3 np+ n2p (ln n)2p+ (ln n)3p 1
Department: Id number: Name:
3. (12 pts.) Determine whether the series is convergent or divergent. Give reasons for your answers.
∞
∑
n=0
enπ
πne
4. (12 pts.) Determine whether the series is convergent or divergent. Give reasons for your answers.
∞ ∑ n=1 ( 1− cos(n2+ 1))2 n2(e1/n− 1)6/7 2
Department: Id number: Name:
5. (12 pts.) Determine whether the series is convergent or divergent. Give reasons for your answers. We denote by⌈ab⌉ the smallest integer bigger than or equal to ab.
∞ ∑ n=1 (−1)⌈n4⌉+1 n = 1 1+ 1 2+ 1 3+ 1 4− 1 5− 1 6− 1 7− 1 8+ 1 9+· · ·
6. (16 pts.) Determine the interval of convergence for the following power series.
(1) (8 pts.) ∞ ∑ n=1 (csch n)xn (2) (8 pts.) ∞ ∑ n=1 ( tan−1(√1 n) ) xn 3
Department: Id number: Name: 7. (12 pts.) Let f (x) = ∫ x 0 ln(1 + 2t2)dt. Then find f(99)(0).
8. (12 pts.) For each t∈ [0, π], let P (t) be the parallelepiped determined by the following three vectors:
u(t) = (2 + sin t, 3 + cos t, 0) v(t) = (2 + sin t, 0, 3 + cos t) w(t) = (3− cos t, 0, 0)
Then find the area of the boundary of P (t) when the volume of P (t) is minimum.
