{"id":6544,"date":"2021-09-30T13:49:43","date_gmt":"2021-09-30T08:19:43","guid":{"rendered":"https:\/\/www.goseeko.com\/blog\/?p=5842"},"modified":"2021-09-30T13:49:43","modified_gmt":"2021-09-30T08:19:43","slug":"what-is-the-ratio-test","status":"publish","type":"post","link":"https:\/\/www.goseeko.com\/blog\/what-is-the-ratio-test\/","title":{"rendered":"What is the ratio test?"},"content":{"rendered":"\n

Overview<\/strong><\/h2>\n\n\n\n

The ratio test<\/a> is also known as D Alembert\u2019s ratio test. It was first published by Jean le Rond d\u2019 Alembert.<\/p>\n\n\n\n

It is sometimes called the Cauchy ratio test. This test is the criterion for the convergence of a series.<\/p>\n\n\n\n

By using this test we check the convergence of any series. If the limit of the ratio between consecutive terms is less than 1 then we can say that the series is divergent.<\/p>\n\n\n\n

Definition of sequence<\/strong><\/h2>\n\n\n\n

A sequence<\/a> is a function from the domain set of natural numbers N to any set S.<\/p>\n\n\n\n

The real sequence is a function from N to R, the set of real numbers; denoted by f: N \u2192 R. Thus the real sequence f is set of all ordered pairs {n, f (n)}|{n = 1, 2, 3, . . .} i.e., set of all pairs (n, f (n)) with n a positive integer.<\/p>\n\n\n\n

Note- Infinite sequence is a sequence in which the number of terms is infinite<\/strong><\/p>\n\n\n\n

Ratio test<\/a><\/h2>\n\n\n\n

Statement- suppose is a series of positive terms\u00a0 such that then,<\/p>\n\n\n\n

1. If k<1 , the series will be convergent.<\/p>\n\n\n\n

2. If k>1 , then the series will be divergent.<\/p>\n\n\n\n

3. The test fails when k = 1<\/p>\n\n\n\n

Solved examples<\/h2>\n\n\n\n

Example-1: Test the convergence of the series whose n\u2019th term is given below-<\/p>\n\n\n\n

n\u2019th term = <\/p>\n\n\n\n

sol. <\/p>\n\n\n\n

We have<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

By D\u2019Alembert ratio test,<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

So that by D\u2019Alembert test, the series will be convergent.<\/p>\n\n\n\n

Example: test the convergence of the following series whose nth term is<\/strong><\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

Sol:<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

Hence the ratio test fails.<\/p>\n\n\n\n

Now using comparison test,<\/p>\n\n\n\n

Since <\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

Since is a harmonic series (with p = 2) which is convergent, so is the series <\/p>\n\n\n\n

Example: test the convergence of the following series whose nth term is<\/strong><\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

Solution:<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

So the series is convergent.<\/p>\n\n\n\n

Interested in learning about similar topics? Here are a few hand-picked blogs for you!<\/p>\n\n\n\n