Math 166 Exam Review
Chapter 8 & Sections 9.1-9.5
Supplemental Instruction
Iowa State University
Leader:
Course:
Instructor:
Date:
Becca
Math 166
Dr. Martin
11/07/12
1) Circle either True or False. You don’t need to explain your answer:
𝑎𝑛+1
𝑛→∞ 𝑎𝑛
a) Ratio Test tells us that a series converges for 𝜌 > 1 𝑤ℎ𝑒𝑟𝑒 𝜌 = lim
TRUE
FALSE
b) A series is a sum of partial sequences.
TRUE
FALSE
1
c) A p-series, ∑∞
𝑘=1 𝑘 𝑝 converges for p > 1.
TRUE
FALSE
d) Geometric Series always have a sum.
TRUE
FALSE
e) The nth term test is most helpful for divergence. We use the direction of the Theorem that states
" 𝐼𝑓 lim 𝑎𝑛 ≠ 0 𝑡ℎ𝑒𝑛 ∑ 𝑎𝑛 𝑑𝑖𝑣𝑒𝑟𝑔𝑒𝑠".
𝑛→∞
TRUE
FALSE
f) If ∑ 𝑎𝑛 converges, but ∑ |𝑎𝑛 | diverges, then we say ∑ 𝑎𝑛 is absolutely convergent.
TRUE
FALSE
1060 Hixson-Lied Student Success Center  515-294-6624  sistaff@iastate.edu  http://www.si.iastate.edu
Find the following limits. Make sure you have an indeterminate form to use L’Hopital’s Rule. Show
all your work!
2) lim 𝑥 𝑥
𝑥→0
3) lim
sin 𝑥
𝑥→0 𝑒 𝑥
4) lim
2ln(𝑥−2)
𝑥→3 𝑥 2 −2𝑥−3
1
1
𝑥→0 𝑥
𝑥2
5) lim (
4−
)
Find the following improper integrals. Show all your work.
3 5
6) ∫0
𝑥−3
∞
7) ∫1
𝑑𝑥
1
(𝑥−1)2
𝑑𝑥
Determine whether the series converges or diverges and state what test you used.
If you can, find the sum. Show all your work.
8) 0.9191919191 ….
(first write this as an infinite series)
9) ∑∞
𝑘=1
ln 2
𝑘
10)
∑∞
𝑛=1
11)
∑∞
𝑛=1
𝑛+1
10𝑛+12
2𝑛 +3𝑛
4𝑛
12)
1
∑∞
𝑛=1 3
√𝑛
Use the integral test to determine whether the series converges or diverges. State your
hypotheses and show all work.
13)
−𝑛
∑∞
𝑛=1 𝑛𝑒
2
Write down an expression for the nth partial sum of this series. Simplify and determine
whether the series converges or diverges. Show all work.
14)
∑∞
𝑘=1 ln (
𝑘
)
𝑘+1
For the following problems, use Limit Comparison Test, Ordinary Comparison Test, Ratio
Test, Absolute Ratio Test, or Alternating Series Test. State the test used and any
hypotheses (when appropriate). Show all work.
15)
∑∞
𝑛=1
16)
∑∞
𝑛=1
17)
∑∞
𝑛=1
18)
∑∞
𝑛=1
1
2𝑛 −1
𝑛−1
𝑛3 +1
2𝑛
𝑛20
(−1)𝑛−1
𝑛2
(−3)𝑛
19)
∑∞
𝑛=1
20)
∑∞
𝑛=1
21)
𝐄𝐬𝐭𝐢𝐦𝐚𝐭𝐞 𝐭𝐡𝐞 𝐞𝐫𝐫𝐨𝐫 𝐟𝐨𝐫 # 𝟏𝟖 𝐚𝐧𝐝 #𝟐𝟎 𝐮𝐬𝐢𝐧𝐠 𝐩𝐚𝐫𝐭𝐢𝐚𝐥 𝐬𝐮𝐦 𝑺𝟓 𝒂𝒔 𝒂𝒏
𝒂𝒑𝒑𝒓𝒐𝒙𝒊𝒎𝒂𝒕𝒊𝒐𝒏 𝒕𝒐 𝒕𝒉𝒆 𝒔𝒖𝒎 𝑺 𝒐𝒇 𝒕𝒉𝒆 𝒔𝒆𝒓𝒊𝒆𝒔:
𝑛!
(−1)𝑛+1
𝑛!