How do you use the ratio test to test the convergence of the series #∑ (8^n)/(n!)# from n=1 to infinity?

Answer 1

The series would converge.

To test the convergence apply the ratio test which means to find

#lim_ (n->oo) (a_(n+1) /a_n)#. If limit is <1, the series would converge. In the present case this limit is,
=# lim_ (n->oo) (8^(n+1))/((n+1)!) * (n!)/8^n#
=# lim_ (n->oo) 8/(n+1)# =0.Hence the series would converge.
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{8^n}{n!}) from (n=1) to infinity, follow these steps:

  1. Compute the ratio of consecutive terms of the series: [ \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = \lim_{n \to \infty} \left| \frac{\frac{8^{n+1}}{(n+1)!}}{\frac{8^n}{n!}} \right| ]

  2. Simplify the ratio: [ = \lim_{n \to \infty} \left| \frac{8^{n+1} \cdot n!}{8^n \cdot (n+1)!} \right| ]

  3. Cancel common terms: [ = \lim_{n \to \infty} \left| \frac{8}{n+1} \right| ]

  4. Evaluate the limit: [ = 0 ]

  5. Analyze the limit:

    • If the limit is less than 1, then the series converges absolutely.
    • If the limit is greater than 1 or infinite, then the series diverges.
    • If the limit equals 1, the test is inconclusive.

Since the limit is 0, which is less than 1, by the ratio test, the series (\sum \frac{8^n}{n!}) converges absolutely.

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