Question

Does sequence converge or diverge calculator.

A) Check for monotonicity
B) Apply the ratio test
C) Use the Cauchy criterion
D) Find the limit

Answer 1

To determine if a sequence converges or diverges, you can follow these steps: check for monotonicity, apply the ratio test, use the Cauchy criterion, and find the limit.

Step-by-step explanation:

In order to determine if a sequence converges or diverges, you can follow these steps:

A) Check for monotonicity:

1. If the sequence is increasing or decreasing, it is called monotonic. If it is non-increasing or non-decreasing, it is called non-monotonic.
2. If the sequence is monotonic and bounded, it converges. Otherwise, it diverges.

B) Apply the ratio test:

1. Calculate the limit as n approaches infinity of the absolute value of the ratio of the (n+1)-th term to the n-th term.
2. If the limit is less than 1, the sequence converges. If it is greater than 1, the sequence diverges. If it is equal to 1, the test is inconclusive.

C) Use the Cauchy criterion:

1. Check if the fraction |a_{n+m} - a_n| is less than epsilon for all positive integers n and m, where epsilon is a positive number.
2. If the fraction satisfies the condition for any positive epsilon, the sequence converges. Otherwise, it diverges.

D) Find the limit:

1. If the sequence passes the above tests and converges, find its limit by evaluating the terms as n approaches infinity.

By following these steps, you can determine if .

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