Ratio Test for Series: Statement, Solved Examples

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 n=1an be a series of positive real numbers. Suppose that

ℓ = limn→∞ an+1an.

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

Q1: Test the convergence of 2+222+233+.

Answer:

The given series = n=12nn

Here, an=2nn.

Therefore,

ℓ = limn→∞ an+1an

= limn→∞ (2n+1n+1×2nn)

= 2 limn→∞ nn+1

= 2 limn→∞ nnn+1n

= 2 limn→∞ 11+1n

= 2 × 11+0

= 2.

As this limit is greater than 1, by ratio test we conclude that the given series diverges.

Q2: Test the convergence of 1+32!+53!+74!+.

Answer:

The given series = n=12n1n!

Here, an=2n1n!.

Therefore, limn→∞ an+1an

= limn→∞ (2(n+1)1(n+1)!×2n1n!)

= limn→∞ (2n+1(n+1)(2n1))

= 0.

As this limit is less than 1, by ratio test, the given series converges.

Q3: Test the convergence of 11!+222!+333!+.

Answer:

The given series = n=1nnn!.

So an = nnn!.

Thus, ℓ = limn→∞ an+1an

= limn→∞ ((n+1)n+1(n+1)!×n!nn)

= limn→∞ (n+1n)n

= limn→∞ (1+1n)n

= 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.

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Q4: Find the values of x, for which the series x+x22+x33+ is convergent or divergent.

Answer:

The given series = n=1xnn

Thus an = xnn.

Now, ℓ = limn→∞ an+1an

= limn→∞ (xn+1n+1×nxn)

= x× limn→∞ nn+1

= x×1=x.

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 1+12+13+ =n=11n. It is a p-series with p=1. Thus, the series diverges.

Final conclusion: The series ∑xn/n! converges if x<1 and diverges if x≥1.

Test the convergence of the following series:

  1. n=11n2n
  2. 1+11!+222!+333!+ [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.

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