Using the Ratio Test for Series Convergence

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  • 0:04 Series & Summation Notation
  • 2:15 The Ratio Test
  • 4:48 Graphical View of Convergence
  • 5:20 Lesson Summary
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Lesson Transcript
Instructor: Gerald Lemay

Gerald has taught engineering, math and science and has a doctorate in electrical engineering.

In this lesson we explore the idea of convergence and how using the ratio test can be useful. As an example, we show how the ratio test can predict the convergence of a particular series.

Series and Summation Notation

As with most technical discussions, the topic of ''series convergence'' has specialized terminology and definitions.


Writing a series with the equation appearing here, we'll define k as the lower limit of a sum, while an is called the general term. Is there an upper limit on the sum? Sure, the upper limit is n = infinity (written as ∞).

Let's use an example to clarify what's going on.


In this example, we can identify the k and the an. The k is 1 and


The capital sigma symbol you've been seeing in these formulas (Σ) is summation notation, which means to sum the general term from n = 1 to n = ∞. To make sense out of this, let's do a few sums.

Summing from n = 0 to n = 2, we get the formulas appearing here:


But 21 is 2 and 22 is 4. Thus, we get:


Therefore, summing the first two terms gives us a result of 1.

What happens if we sum a little further? Let's try summing up to n = 5:


Simplifying, we get:


So, from a sum equal to 1 using the first two terms, we now have a sum equal to ≅ 1.78.

We could let M be the upper value for n. When we summed up to M = 2, we got a result of 1. When we summed up to M = 5, the result was 57/32. The idea of convergence is intuitive: if the series converges, the sum settles in to a finite value as M gets larger and larger. By the way, not all series will converge.

We could experiment by summing the series for increasing values of M. This would be a ''brute force'' approach. Intuitive but time-consuming and not very elegant. Instead, we can use a convergence test for determining convergence. There are many such tests. In this lesson, we explore convergence using the ratio test.

The Ratio Test

The ratio test instructs us to find a limit we call ''L'', as you can see in this formula appearing here:


There are three instructions implied:

  1. form a ratio of an+1 divided by an
  2. take the absolute value of this ratio
  3. take the limit as n → ∞

Once we find L, we then conclude that:

  • if L < 1, the series converges
  • if L > 1, the series does not converge; it diverges
  • if L = 1, the test is inconclusive and we can't tell from this test if the series converges or not

We can break down ratio testing into steps.

Step 1: Identify the general term, an.

In our example,


Step 2: Find an expression for an+1.

Simply replace the n in an with n+1:


Step 3: Build the ratio of an+1 to an.


Step 4: Simplify the ratio.

First, note division of fractions is pretty cumbersome. We can write division by a fraction as multiplication with the reciprocal of the fraction, like you can see here:


Then, organize the terms.


See how the ''2-to-the-power terms'' are written one over the other? Same for the n terms.

Further simplifying, we get the equation appearing here:


2n and 2n+1 have the same base of 2. This allows us to combine the exponents and express the division as a subtraction of powers: n + 1 - n.

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