Simple limit of sequences question.

In summary, the question is whether the series {An} = 2n / n! is convergent or divergent. The person asking for help initially tried dividing both terms by n! to get 0, but then considered a different approach of dividing one term by the next term which led to the conclusion that the limit goes to 0, indicating that the series is convergent.
  • #1
nothingkwt
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0
Let {An} = 2^n / n!

is it convergent or divergent and why?
 
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  • #2
Not right forum.
No effort shown.
 
  • #3
Any suggestions yourself?
 
  • #4
nothingkwt said:
Let {An} = 2n / n!

is it convergent or divergent and why?

(try using the X2 and X2 buttons just above the Rreply box :wink:)

tell us what you think, and why, and then we'll comment! :smile:

(and please use the homework forum in future)
 
  • #5
Sorry I am new to this forum.

I thought of dividing both terms with the term that approached infinity more rapidly which was n!. which gave me 0, but then I saw that if 2n was actually 10n for example it would approach infinity more rapidly which would give me ∞ but I'm not entirely sure. So I got confused.
 
  • #6
nothingkwt said:
I thought of dividing both terms with the term …

try dividing one term by the next term instead :smile:

(ie An/An+1)
 
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  • #7
2*2*2*2*2... / n(n-1)(n-2)(n-3)...

So the limit goes to 0 it's clear now thanks.
 

1. What is a sequence in mathematics?

A sequence is a list of numbers arranged in a specific order. Each number in the sequence is called a term, and the position of the term in the sequence is known as its index. Sequences are commonly used in mathematics to represent patterns and relationships between numbers.

2. What is a limit of a sequence?

The limit of a sequence is the value that the terms of the sequence approach as the index approaches infinity. In other words, it is the value that the terms of the sequence get closer and closer to, but may never actually reach.

3. How do you find the limit of a sequence?

To find the limit of a sequence, you can use the following steps:1. Write out the terms of the sequence.2. Observe the pattern of the terms.3. If the pattern is obvious, you can simply state the limit.4. If the pattern is not obvious, you can use the limit definition to evaluate the limit.5. If the limit exists, you can state its value. If the limit does not exist, you can state that the sequence has no limit.

4. What is the difference between a bounded and an unbounded sequence?

A bounded sequence is a sequence in which all the terms are within a certain range or bound. In other words, there is a finite number that is greater than or equal to all terms in the sequence, and a finite number that is less than or equal to all terms in the sequence. An unbounded sequence, on the other hand, does not have this property and can have terms that approach infinity or negative infinity.

5. Why is finding the limit of a sequence important?

Finding the limit of a sequence is important because it allows us to understand the behavior of the sequence as the index approaches infinity. It can also help us determine the convergence or divergence of a sequence, which has practical applications in fields such as calculus, statistics, and physics. Additionally, understanding the limit of a sequence is essential for understanding more advanced mathematical concepts such as series and continuity.

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