How do you use the ratio test to test the convergence of the series #∑(n!)/(n^n)# from n=1 to infinity?

Answer 1

The series converges.

The term of the series is

#a_n=(n!)/(n^n)#

The ratio test is

#|a_(n+1)/a_n|=|((n+1)!)/((n+1)^(n+1))*n^n/(n!)|#
#=|((n+1))/((n+1)^(n+1))*n^n|#
#=|(n^n)/(n+1)^n|#

Therefore,

#lim_(n->oo)|a_(n+1)/a_n|=lim_(n->oo)|(n^n)/(n+1)^n|#
#=lim_(n->oo)(n/(n+1))^n#
#=lim_(n->oo)((n+1)/(n))^-n#
#=1/e#

As

#lim_(n->oo)|a_(n+1)/a_n| <1#

The series converges.

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 2

To use the ratio test to test the convergence of the series (\sum \frac{n!}{n^n}) from (n=1) to infinity, follow these steps:

  1. Calculate the limit of the ratio (L) as (n) approaches infinity: [L = \lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right|] where (a_n = \frac{n!}{n^n}).

  2. Compute the ratio of consecutive terms: [\frac{a_{n+1}}{a_n} = \frac{(n+1)!}{(n+1)^{n+1}} \cdot \frac{n^n}{n!}] [= \frac{(n+1)!}{n!} \cdot \frac{n^n}{(n+1)^{n+1}}] [= \frac{n+1}{(1+\frac{1}{n})^n}]

  3. Take the limit as (n) approaches infinity: [L = \lim_{n \to \infty} \frac{n+1}{(1+\frac{1}{n})^n}]

  4. Determine the convergence of the series based on the value of (L):

    • If (L < 1), the series converges absolutely.
    • If (L > 1) or (L = \infty), the series diverges.
    • If (L = 1), the ratio test is inconclusive, and other tests may be needed to determine convergence.
  5. Therefore, to test the convergence of the series (\sum \frac{n!}{n^n}), calculate the limit (L) and compare it to 1 to make the determination.

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer from HIX Tutor

When evaluating a one-sided 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 one-sided 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 one-sided 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 one-sided 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.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7