A Closer Look at the (1+1/n)^n Limit

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Homework Help Overview

The discussion revolves around the limit of the expression (1 + 1/n)^n as n approaches infinity, a topic commonly associated with calculus and the concept of exponential growth.

Discussion Character

  • Exploratory, Conceptual clarification

Approaches and Questions Raised

  • Participants explore the limit of the expression and question whether the limit can be taken inside the brackets. There is a mention of the ratio test being inconclusive, and some participants reference the standard limit that leads to the number e.

Discussion Status

The discussion is active, with participants seeking clarification on the steps leading to the conclusion that the limit approaches e. Some guidance has been offered regarding the use of the binomial theorem for expansion, but there is no explicit consensus on the approach yet.

Contextual Notes

There is an indication of uncertainty regarding the application of limits and the validity of the ratio test in this context. The original poster expresses confusion about how to proceed with the problem.

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Homework Statement


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The Attempt at a Solution



I came up with this answer and i don't know how to continue:

(1+1/n)^n

I understand I have to put in limit n -> infinty. But can i put the limit inside the bracket? So i will get an answer of 1 ?

So ultimately, the ratio test is inconclusive?
 
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(1 + (1/n))n → e (≈ 2.718) as n → ∞.
 
How did you get that? Can you bring me through the steps?
 
It's a standard limit. Use the binomial theorem to expand (1 + (1/n))n, and then take the limit n → ∞. You should get

1 + 1 + 1/2! + 1/3! + ... = e
 

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