MATH214 Study Guide - Quiz Guide: Alternating Series Test, Ratio Test, Alternating Series
SchoolUniversity of Alberta
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Problem 1. Is the following series convergent? Is it absolutely convergent? Is it conditionally
Problem 2. Find the radius and interval of convergence of the following series:
Explain your solution! Write the answer! Answer is not written – problem is not solved!
Problem 1. The general term is an=(−1)nln2(n)
n. This is an alternating series. To use the alternat-
ing series test, we need to check that ln2(n)
nis decreasing and goes to zero. Clearly lim
Now if we calculate the derivative of f(x) = ln2(x)
xwe get f0(x) = ln(x)·(2−ln(x))
x2which is negative
whenever x>e2. Therefore for n>9, ln2(n)
nis decreasing. By the alternating series test, we deduce
that Pn>9anis convergent. Since the ﬁrst 8 terms won’t aﬀect the convergence we deduce that the
series is convergent.
Now to check if the series is absolutely convergent, we need to consider the series of the absolute
values. We have |an|=ln2(n)
n. When n>3, we have ln2(n)
n. We know that
(Harmonic series), then by the comparison test we have that
|an|is divergent. This means that
anis not absolutely convergent.
Since our series is convergent but not absolutely convergent, it is conditionally convergent.
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