Finding the Limit of a Strange Sequence: How to Use Stirling's Approximation

In summary, a limit of a strange sequence is a value that the terms of the sequence approach as the number of terms approaches infinity. It is typically calculated by examining the pattern of the terms in the sequence and determining the value that the terms approach as the sequence continues infinitely. If a strange sequence does not have a limit, it means that the terms do not approach a specific value as the number of terms increases. The concept of a limit is important in mathematics because it allows us to understand the behavior of a sequence or function at a particular point or as it approaches infinity, and it is a crucial concept in calculus and other mathematical disciplines. A strange sequence can only have one limit, and if it has multiple limits, it is considered to be
  • #1
JJ6
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Homework Statement



I need to find the limit of the sequence:

[(1+1/n)(1+2/n)(1+3/n)...(1+n/n)]^(1/n) as n approaches infinity.

I know that the limit should come out to 4/e, but I cannot figure out why.

Homework Equations



None.

The Attempt at a Solution



The original sequence is equivalent to [(2n)!/(n!*n^n)]^(1/n), but I have absolutely no idea what to do with it after that. Can anybody point me in the right direction?
 
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  • #2
Why not just apply Stirling's approximation to the original sequence?
 

FAQ: Finding the Limit of a Strange Sequence: How to Use Stirling's Approximation

What is a limit of a strange sequence?

A limit of a strange sequence is a value that the terms of the sequence approach as the number of terms approaches infinity. It is the ultimate behavior of the sequence.

How is the limit of a strange sequence calculated?

The limit of a strange sequence is typically calculated by examining the pattern of the terms in the sequence and determining the value that the terms approach as the sequence continues infinitely. This can also be calculated using mathematical formulas and techniques.

What does it mean if a strange sequence does not have a limit?

If a strange sequence does not have a limit, it means that the terms do not approach a specific value as the number of terms increases. Instead, the terms may oscillate or diverge, meaning they do not follow a distinct pattern.

Why is the concept of a limit important in mathematics?

The concept of a limit is important in mathematics because it allows us to understand the behavior of a sequence or function at a particular point or as it approaches infinity. It is also a crucial concept in calculus and other mathematical disciplines.

Can a strange sequence have multiple limits?

No, a strange sequence can only have one limit. If a sequence has multiple limits, it is considered to be undefined or divergent. A sequence cannot approach more than one value as the number of terms increases.

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