Science:Infinite Series Module/Units/Unit 2/2.4 The Ratio Test/2.4.01 Introduction to The Ratio Test

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In previous lessons, we explored convergence tests that applied to

  • series with positive terms (the integral test)
  • series with alternating terms (the alternating series test)

However, there are series for which the above tests cannot be applied. We could ask: how would we determine if an infinite series converges or diverges if its terms irregularly switch from positive to negative?


Consider for example the recursive sequence

Failed to parse (syntax error): {\displaystyle a_1 = 1, \qquad a_{k+1} = \frac{0.1 + \cos(k)}{\sqrt{k}}a_k  }

for k = 1, 2, 3, ...

Approximate values of the first eight terms are given in the table below.

k ak
1 1.00000
2 -0.22355
3 +0.11487
4 -0.03180
5 -0.00546
6 -0.00236
7 -0.00076
8 +0.00001

We cannot use the integral test (not all terms are positive).

We cannot use the alternating series test (the signs change irregularly).

We will see that we can use the ratio test to show that this series converges.