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### Course: AP®︎/College Calculus BC>Unit 10

Lesson 8: Ratio test for convergence

# Ratio test

The ratio test is a most useful test for series convergence. It caries over intuition from geometric series to more general series. Learn more about it here.

## Want to join the conversation?

• You can't prove that n^10/n! doesn't diverge with the divergence test as stated in because lim an = 0 so it can still diverge ou converge. Or am I wrong?
• Absolutely correct, a series such as 1/n (harmonic series) diverges, even though the limit as n goes to infinity leads to 0 for an, so Sal made a mistake.
• So I am currently taking BC Calculus and truly appreciate the help Khan Academy has been giving. I am learning about Series and Sequences and can't seem to find a video on the Root Test. I might not have seen it and would appreciate if anyone can guide me to a video or website that could help me with this test.
• The same conclusions apply as with this test. The only difference is that you take the limit of the absolute value of the nth root of the function.
• Why did you keep (n+1)^10/(n+1) instead of just making (n+1)^10 (n+1)^9?
• There's no particular reason. At that point it's pretty easy to see the solution either way, and Sal apparently didn't feel the need to factor out (n+1).
• why can't we conclude using this test whether the series converges or diverges when the lim of the ratio as n->∞ = 1 ?
• The way the ratio test works is by evaluating the absolute value of the ratio when applied after a very large number of times (tending to infinity), regardless of the initial terms in the series.

If the ratio near infinity is less than 1, then we know for certain that each term is becoming less and less and the series will converge. If the ratio near infinity is greater than 1, then we know that each term will continue to grow, so the series will diverge.

But if the ratio near infinity is equal to 1, then the ratio will not cause the initial terms of the series to either grow or become smaller, so the convergence or divergence of the series cannot be determined by the ratio, additional information about the first terms of the series is needed to decide. This is why when the ratio test give a 1, the results are inconclusive.
• Does this work for only geometric series?
• I think it works on all series, as Sal used it on a series that wasn't a geometric series.
• So I understand the ratio and divergence test, but I don't quite know when to use the tests? Is there something I should look for to signify which test to use?
• Unlike Ratio test, you cannot determine if a series is convergent from the divergent test. Even if the divergent test fails . it does not mean the series is convergent( eg: take the series sigma 1/n).
I would start with the ratio test, because it seems more definitive.
• What is a factorial "!"?
• If L is equal to 1 why is it inconclusive ? Don't we know that a series which does not converge ,diverge . So if common ratio is approaching 1 then it should diverge as there is no in between .
Also how do we know tht n^10/n! is in geometric progression ,or this test applies to any series other than G.P?
~thankyou
• 1 / n diverges, with L = 1; 1 / n^2 converges, also with L = 1. Just with these two examples, we have shown that when L = 1, we cannot be sure of convergence or divergence.

n^10 / n! is definitely not geometric, but the ratio test applies to all series. The geometric series test is just a specific case of the ratio test.
• limit n->inf: (25^n)/(2^n^2).
(25^n+1)/(2^(n+1)^2) * (2^n^2)/(25^n) = (25*25^n)/((2^(n+1)^2) * (2^n^2)/(25^n)
= 25* [ (2^n)/(2*2^n) ]^2 =25*(1/2)^2 = 25/4

My online homework says that this is incorrect. Where did I go wrong?