ICOEE 2018

Zopam Anry

In the world of mathematics, series convergence plays a crucial role in determining whether a mathematical series converges to a specific value or diverges into infinity. Two powerful tools in this realm are the Ratio Test and the Root Test. In this article, we will delve into these tests, understand their principles, and learn how to apply them effectively.

The Ratio Test

The Ratio Test is a fundamental tool for examining the convergence of a series. It states that for a given series ∑(a_n), if the limit of the absolute value of the ratio of consecutive terms exists and is less than 1, then the series is absolutely convergent.

Mathematically, the Ratio Test can be expressed as:

lim (n→∞) |a_(n+1) / a_n| < 1

Here’s a step-by-step guide on how to apply the Ratio Test:

  1. Compute the limit lim (n→∞) |a_(n+1) / a_n|.
  2. If the limit is less than 1, the series is absolutely convergent.
  3. If the limit is greater than 1 or does not exist, the series is divergent.
  4. If the limit is equal to 1, the test is inconclusive, and you may need to employ other convergence tests.

Example using the Ratio Test

Let’s say we have the series ∑(1/n!). To determine its convergence using the Ratio Test:

  1. Calculate the limit:
lim (n→∞) |(1/(n+1)!)/(1/n!)| = lim (n→∞) |1/(n+1)| = 0
  1. Since the limit is less than 1, the series ∑(1/n!) is absolutely convergent.

The Root Test

The Root Test is another valuable tool for assessing the convergence of a series. It states that for a given series ∑(a_n), if the limit of the nth root of the absolute value of the terms exists and is less than 1, then the series is absolutely convergent.

Mathematically, the Root Test can be expressed as:

lim (n→∞) (|a_n|)^(1/n) < 1

Here’s how to apply the Root Test:

  1. Compute the limit lim (n→∞) (|a_n|)^(1/n).
  2. If the limit is less than 1, the series is absolutely convergent.
  3. If the limit is greater than 1 or does not exist, the series is divergent.
  4. If the limit is equal to 1, the test is inconclusive, and you may need to explore other convergence tests.

Example using the Root Test

Consider the series ∑(1/n^2). To determine its convergence using the Root Test:

  1. Calculate the limit:
lim (n→∞) (|1/n^2|)^(1/n) = lim (n→∞) 1/n = 0
  1. Since the limit is less than 1, the series ∑(1/n^2) is absolutely convergent.

Summary

The Ratio Test and Root Test are essential tools in the study of series convergence. By applying these tests, mathematicians and scientists can determine whether a series converges to a finite value or diverges into infinity. Remember that these tests provide valuable insights into the behavior of mathematical series, aiding in various fields of mathematics and science.

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