How do you use the ratio test to test the convergence of the series #sum_(n=1)^oo (n!)/((2n+1)!)#?
The series converges
Then, the ratio test is
Therefore,
By signing up, you agree to our Terms of Service and Privacy Policy
To use the ratio test to test the convergence of the series ( \sum_{n=1}^{\infty} \frac{n!}{(2n+1)!} ), follow these steps:

Write down the general term of the series, which is ( a_n = \frac{n!}{(2n+1)!} ).

Apply the ratio test, which states that if ( \lim_{n \to \infty} \left \frac{a_{n+1}}{a_n} \right ) exists, then:
 If the limit is less than 1, the series converges.
 If the limit is greater than 1 or the limit is infinity, the series diverges.
 If the limit is equal to 1, the ratio test is inconclusive.

Find ( a_{n+1} ) and ( a_n ):
 ( a_{n+1} = \frac{(n+1)!}{(2(n+1)+1)!} = \frac{(n+1)!}{(2n+3)!} )
 ( a_n = \frac{n!}{(2n+1)!} )

Take the ratio ( \frac{a_{n+1}}{a_n} ) and simplify:
 ( \frac{a_{n+1}}{a_n} = \frac{(n+1)!}{(2n+3)!} \cdot \frac{(2n+1)!}{n!} = \frac{n+1}{(2n+2)(2n+3)} )

Calculate the limit as ( n \to \infty ) of the absolute value of the ratio:
 ( \lim_{n \to \infty} \left \frac{a_{n+1}}{a_n} \right = \lim_{n \to \infty} \frac{n+1}{(2n+2)(2n+3)} )

Simplify and find the limit:
 ( = \lim_{n \to \infty} \frac{1}{4} \cdot \frac{n+1}{n+\frac{3}{2}} = \frac{1}{4} )

Since ( \frac{1}{4} < 1 ), the series converges by the ratio test.
By signing up, you agree to our Terms of Service and Privacy Policy
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
 How do you use the Nth term test on the infinite series #sum_(n=1)^oo(n(n+2))/(n+3)^2# ?
 How do you determine if #a_n=11.1+1.111.111+1.1111...# converge and find the sums when they exist?
 How do I use the Limit Comparison Test on the series #sum_(n=1)^oosin(1/n)# ?
 How do you find the positive values of p for which #Sigma lnn/n^p# from #[2,oo)# converges?
 Why does the integral test not apply to #Sigma (1)^n/n# from #[1,oo)#?
 98% accuracy study help
 Covers math, physics, chemistry, biology, and more
 Stepbystep, indepth guides
 Readily available 24/7