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Comparison tests

# Direct comparison test

AP.CALC:
LIM‑7 (EU)
,
LIM‑7.A (LO)
,
LIM‑7.A.8 (EK)
If every term in one series is less than the corresponding term in some convergent series, it must converge as well. This notion is at the basis of the direct convergence test. Learn more about it here.

## Want to join the conversation?

• I don't understand why the sequence can't oscillate just because its terms are all non-negative.
Can someone please explain why? I mean, for example, can't a sequence be like 3,4,3,4,3,4,3,4,3,4,3,4,... which is oscillating between two values but has all non-negative values??
• From the author:In order for the sequence of partial sums to oscillate, you would need negative terms in the series. For example, to go from 4 to 3, you'd have to add -1. We made the assumption that every term in the sum is non-negative.
• , I don't quite understand how we can define that `Σ a_n` is "smaller" than `Σ b_n`? We compare the summations between them or what?
• One of the assumptions established at the beginning is that each term in the "a" sequence is less than or equal to the corresponding term in the "b" sequence (that is, a_n is less than or equal to b_n). As a result, the sum of the terms in the "a" sequence has to be less than or equal to the sum of the terms in the "b" sequence.
• Hello there,
If one series is convergent, the another is also convergent.
If one series is divergent, the other one is also divergent.
or
If one greater series is convergent, the another is also convergent.
If one smaller series is divergent, the other one is also divergent.

Which one do you want to mean in the video? I don't need to care about which is bigger or smaller, do I?

Thanks
Aung
• The second version you said is correct.

We say a series diverges if it adds all the way to infinity, right? It makes sense that if there's a series that diverges, a series larger than that one will also diverge. This happens because the smaller one went to infinity, so of course the bigger one will too. Similar thinking goes for convergent series. Convergence is defined as a series that adds up to a finite number. So if there's a series smaller than one that converges to a finite number, it wouldn't make sense for the smaller one to all the way up to infinity because a series larger than it converged.

Hope that helped.
• what if b(n) diverges , will that make a(n) diverges.
thanks
• Since b(n) is ≥ a(n) for all n, if b(n) diverges, it says nothing about a(n) and the divergence test is not applicable.

This is what you must understand about the divergence test.....
If you have two different series, and one is ALWAYS smaller than the other, THEN
1) IF the smaller series diverges, THEN the larger series MUST ALSO diverge.
2) IF the larger series converges, THEN the smaller series MUST ALSO converge.

You should rewatch the video and spend some time thinking why this MUST be so. Understanding this is paramount to moving forward with respect to understanding the divergence and convergence of series.

Keep Studying and Keep Asking Questions!
• why does a_n,b_n have to be >= 0? Why can not both be nagative, just not equal to 0?
• At , Sal writes a[n],b[n] <=0.
Is this just a given part of the problem i.e. is he saying "Imagine two series, a[n] and b[n] where both of them are >=0..."
OR, did Sal use some information about the series to determine that they are nonnegative?
• Sal is defining the properties of a[n] and b[n] so that the comparison test is valid and so that you can see the what is required of the series you wish to test in order to apply the "Comparison Test" - so your first thought is correct although this isn't a problem, it is (the basis of) a theorem.
• We always skip the case when the sequence contains negative numbers.
So what test (or method) can be applied when we examine a sequence with negative numbers??
for example,
a sequence like,
1, -0.1, 0.01, -0.001, 0.0001.........
its series converge to 0.9090909090..................a repeating decimal.

do we have a rule for them?
(1 vote)
• What's the intuition behind why if the smaller series diverges then the larger series diverges? Why not if the larger series diverges then the smaller series diverges too? And I have the same questions except regarding convergence instead of divergence.
(1 vote)
• If 𝑥 > 𝑦 and 𝑦 > 𝑧, it follows that 𝑥 > 𝑧. So if we consider divergence to be approaching ∞, then if a series "diverges", and the partial sums of another series are larger than the original divergent series, then it follows that larger series must also be divergent.

Using our previous inequalities, if we are given that the infinite series 𝑦 + 𝑦 + 𝑦 + ... diverges, then 𝑥 + 𝑥 + 𝑥 + ... also diverges since it is "larger than something that is already shown to be infinite" loosely speaking.

Comment if you have questions!