Quiz 7
Math 1220–7
November 9, 2012
Key
Directions: Show all work for full credit. Clearly indicate all answers. Simplify all mathematical expressions completely. No calculators are allowed on this quiz. Each part of each
question is worth 15 points.
1. Determine whether each of the following series converges or diverges. Be sure to justify
each answer with an appropriate test.
∞
X
2
(a)
ke−3k (#11 from 9.3)
n=1
Using the integral test, we get:
∞
Z ∞
1
1
1
1 −3x2 −3x2
= − · 0 + e−3 = 3
xe
dx = − e
6
6
6
6e
1
1
Since this integral converges, the sum also converges.
(b)
∞
X
8n
n=1
n!
(#5 from 9.4)
Using the ratio test, we see that
an+1
8n+1 /(n + 1)!
8
=
=
n
an
8 /n!
n+1
Taking the limit as n → ∞, this goes to 0. Since 0 < 1, the sum converges.
(c)
∞
X
n=1
n
(#11 from 9.4)
n + 200
n
= 1 6= 0, the sum diverges by the nth -term test for divergence.
n→∞ n + 200
Since lim
2. Determine whether the following series is absolutely convergent, conditionally convergent, or divergent: (#21 from 9.5)
∞
X
(−1)n+1
n=1
n
n2 + 1
Since this is an alternating series with decreasing terms, and lim
n
n2 +1
∞
X
n
= 0, the series
+1
n
diverges by the Limit Comparison Test (we
+
1
n=1
to 1/n), so the series is conditionally convergent.
converges. However, the series
compare
n→∞ n2
n2