Overview
The ratio test is also known as D Alembert’s ratio test. It was first published by Jean le Rond d’ Alembert.
It is sometimes called the Cauchy ratio test. This test is the criterion for the convergence of a series.
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.
Definition of sequence
A sequence is a function from the domain set of natural numbers N to any set S.
The real sequence is a function from N to R, the set of real numbers; denoted by f: N → 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.
Note- Infinite sequence is a sequence in which the number of terms is infinite
Ratio test
Statement- suppose 
1. If k<1 , the series will be convergent.
2. If k>1 , then the series will be divergent.
3. The test fails when k = 1
Solved examples
Example-1: Test the convergence of the series whose n’th term is given below-
n’th term =
sol.
We have
By D’Alembert ratio test,
So that by D’Alembert test, the series will be convergent.
Example: test the convergence of the following series whose nth term is
Sol:
Hence the ratio test fails.
Now using comparison test,
Since
Since 
Example: test the convergence of the following series whose nth term is
Solution:
So the series is convergent.
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