MATH 101 V2A
March 6th – Practice problems
Hints and Solutions
1. (a) List all of the tests that you have learned in class this week.
(b) For each test listed in (a), find an example of a series whose convergence/divergence can be shown
using that test.
Hint: Look in your notes – you shouldn’t need a solution for this!
2. The Integral Test states that if f (x) is a non-negative, decreasing function, then
Z ∞
and only if
f (x)dx converges.
∞
X
f (n) converges if
n=1
1
Prove the Integral Test by generalizing the method used to prove the p -Test.
Hint: Use the method used to prove the p -Test in class to get the inequality
∞
X
n=2
Z
f (n) ≤
∞
f (x)dx ≤
1
∞
X
f (n).
n=1
Use this inequality to conclude the desired result.
3. Prove that (possibly infinite) decimal expressions represent real numbers; that is, prove that
bk bk−1 · · · b1 b0 . a1 a2 a3 · · ·
is a convergent series.
Solution: See the solutions to the practice problems from March 4th.