This is a comprehensive study guide for MATH 009C at the University of California, Riverside based on the class taught by Professor Peter Samuelson for Spring 2024. Covers all chapters and topics, definitions, and all other major ideas covered by the course on the final exam.
MATH 009C Study Guide
UCR MATH 009C Final Exam Study Guide (Spring 2024) with Professor Peter
Samuelson. Created based on the Final Practice Problems released on Canvas.
(Note: click on images to enlarge them)
Final Practice Problems
MATH 09C: Study Guide 1
, Series Tests
Geometric Series Test: Use for series of the form ∑ ar n
converges if ∣r∣ < 1
Alternating Series Test: Use for series of the form ∑ (−1)n bn
converges if bn is
1. positive
2. decreasing
3. limn→∞ bn = 0
Comparison Test: Compare with a known convergent or divergent series
Limit Comparison Test: Compare the limit of the ratio of the series terms with a
known convergent or divergent series
Ratio Test: Use for series with factorials or exponential terms
a
converges if limn→∞ ∣ an+1 ∣ < 1
n
Root Test: Use for series with nth power terms
converges is limn→∞
n
an < 1
Divergence Test: Use to quickly check if the series diverges
if limn→∞ bn =
0, the series diverges
MATH 09C: Study Guide 2
, Absolutely Convergent, Conditionally
Convergent, or Divergent
∞
Absolutely Convergent: A series ∑n=1 an converges absolutely if the series of
∞
absolute values ∑n=1 ∣an ∣converges
WHY? Because the terms an must be getting small enough for both the
absolute and non-absolute sums to converge
∞ ∞
Conditionally Convergent: A series ∑n=1 an converges conditionally if ∑n=1 an
∞
converges, but ∑n=1 ∣an ∣does not converge (diverges)
MATH 09C: Study Guide 3
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