The ratio test for series is useful to check a series is convergent or divergent. Here we consider the limit of the quotient of (n+1)-th and n-th terms of the series, and depending on this limit is greater or less than 1 we decide its convergence. In this article, we study the ratio test of a series along with some solved examples.
Ratio Test Statement
Let
ℓ = limn→∞
Then we have the following.
- If ℓ<1, then the series ∑an converges.
- If ℓ>1, then ∑an diverges.
- If ℓ=1, we cannot conclude.
Ratio Test Solved Examples
The given series =
Here,
Therefore,
ℓ = limn→∞
= limn→∞
= 2 limn→∞
= 2 limn→∞
= 2 limn→∞
= 2 ×
= 2.
As this limit is greater than 1, by ratio test we conclude that the given series diverges.
The given series =
Here,
Therefore, limn→∞
= limn→∞
= limn→∞
= 0.
As this limit is less than 1, by ratio test, the given series converges.
The given series =
So an =
Thus, ℓ = limn→∞
= limn→∞
= limn→∞
= limn→∞
= e.
Note that the value of e lies between 2 and 3. So the limit ℓ is greater than 1. Hence, by ratio test the given series diverges.
Related Articles:
- Comparison Test for Series: Statement, Examples [Limit Form]
- Convergence of a Series: Definition, Formula, Examples
The given series =
Thus an =
Now, ℓ = limn→∞
= limn→∞
=
=
So by ratio test, the series converges if x<1 and diverges if x>1.
The case x=1. For x=1, the given series is equal to
Final conclusion: The series ∑xn/n! converges if x<1 and diverges if x≥1.
Homework:
Test the convergence of the following series:
-
[Ignore the first term].
FAQs
Q1: State ratio test for series.
Answer: The ratio test for series states the following. If limn→∞ (an+1/an) is less than 1, then the series ∑an converges, and if it is greater than 1 then the series diverges.

This article is written by Dr. Tathagata Mandal, Ph.D in Mathematics from IISER Pune (Algebraic Number Theory), Postdocs at IIT Kanpur & ISI Kolkata. Currently, working as an Assistant Prof. at Adamas University. Thank you for visiting the website.