Educator

Ratio Test and Root Test

Main definitions and theorems:

Definitions: Let ∑a_n be a series.

• ∑a_n is Absolutely Convergent means that ∑a_n converges and ∑|a_n| converges.

• ∑a_n is Conditionally Convergent means that ∑a_n converges but ∑|a_n| diverges.

• ∑a_n is Divergent means that ∑a_n diverges and ∑|a_n| diverges.

Theorem: If ∑|a_n| converges, then ∑a_n converges.

Ratio Test: Suppose ∑a_n is a series. Calculate lim and call it L.

• If L < 1, then the series converges absolutely.

• If L > 1, then the series diverges.

• If L = 1, then the Ratio Test doesn’t tell you. The series could converge absolutely, converge conditionally, or diverge.

Root Test: Suppose ∑a_n is a series. Calculate lim and call it L.

• If L < 1, then the series converges absolutely.

• If L > 1, then the series diverges.

• If L = 1, then the Root Test doesn’t tell you. The series could converge absolutely, converge conditionally, or diverge.

Hints and tips:

• Think of an Absolutely Convergent series as being “super-convergent”. It is so strongly convergent that it still converges even if we make everything positive.

• Think of a Conditionally Convergent series as being “just barely convergent”. It only converges because some of the terms are negative.

• Here are some flags that might suggest the Ratio Test:

• (−1)ⁿ , since many other tests require positive terms.

• n!, (2n)!, since factorials cancel nicely in ratios. (Remember that (n + 1)! = (n + 1)n! and [2(n + 1)]! = (2n)!(2n + 1)(2n + 2).)

• constantⁿ (including eⁿ )

• Any kind of power series/Taylor Series (coming later)

• Here are some flags that might suggest that you not use the Ratio Test:

• n, n² (or any polynomial), √n, ln n. All of these have lim = 1, so the Ratio Test fails. But if you have some of this clutter mixed with some good stuff, Ratio is great for clearing the clutter away and leaving you with something good.

• nⁿ , or more generally (a function of n)^{(another function of n)} (more appropriate for Root Test)

(Note that the Ratio Test is based on comparison with geometric series. So things that look geometric, e.g. constantⁿ , make it work. p-series do not.)

• Remember that .

• Here are some flags that might suggest the Root Test:

• (−1)ⁿ , since many other tests require positive terms.

• (a function of n)^{(another function of n)} , e.g. nⁿ , (n − 2)²ⁿ , etc. Not (constant)ⁿ (use geometric series or Ratio Test) or n^constant (use p-series or LCT).

• Here are some flags that might suggest that you not use the Root Test:

• n!, (2n)!, etc. (Use Ratio Test.)

• n² , n, √n. All of these have lim |a_n|^1/n = 1. But if you have some of this mixed with some stuff that looks good for Root Test, then you can use Root.

• Remember that you can write . Then you can often sort out the exponent using L’Hôpital.

• It is probably worth memorizing that lim = e, or even that
  lim .
  

• Also remember that e ≈ 2.7, so e^k > 1 if k is positive and e^k < 1 if k is negative.