Comparison Tests for Positive Term Series ∑ an
In a Nut Shell: One strategy to evaluate the convergence or divergence of an infinite
series is to compare it with a known series that either converges or diverges. i.e. The
harmonic series is known to diverge so it can be used to test divergence. The p-series
and geometric series are candidates to check for either convergence or divergence.
Click here for a reminder of these series.
Two comparison tests, the Comparison Test and the Limit Comparison Test, are
available to test positive term series for convergence or divergence. They provide
no information about the sum of the series.
Strategy: Compare one series, ∑ bn , where it is known to either converge or
diverge with the series to be tested, ∑ an .
Suggestion: For the test pick ∑ bn that converges if you think ∑ an converges.
Likewise, for the test pick ∑ bn that diverges if you think ∑ an diverges.
Comparison Test
Suppose 0 ≤ an ≤ bn for all n; If ∑ bn converges, then ∑ an converges
Suppose 0 ≤ bn ≤ an for all n; If ∑ bn diverges, then ∑ an diverges
Limit Comparison Test (Interested in large values of n)
Suppose an ≥ 0, bn > 0 for all n and
lim ( an / bn ) = L (an and bn are sequences ) and L is a finite number n → ∞ ∞ ∞ If L > 0, then the series ∑ an and ∑ bn either n = 1 n = 1 both series converge or both series diverge.
If L = 0 , an ‘s are much smaller than bn ‘s for large n ∞ ∞ If L = 0, ∑ bn converges, then ∑ an converges. n =1 n = 1
Click here for examples.
Copyright © 2019 Richard C. Coddington
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