Can Anyone Prove why e^x= (1+x/n)^n as n approaches Infinity

Thread starter tam421602
Start date Mar 1, 2010
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E^x Infinity

In summary, the equation e^x = (1+x/n)^n as n approaches Infinity is a representation of the definition of e, which is a mathematical constant. This can be proven using the limit definition of e and it is valid for all values of x. It is commonly used in real life, such as in compound interest calculations.

Mar 1, 2010

tam421602

Can anyone help me ? I am completely lost on this one

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Mar 1, 2010

Sweet_GirL

tam421602 said:

Can anyone help me ? I am completely lost on this one

Hello,
Start by using the substitution:

[tex]m=\frac{n}{x}[/tex]

What did you get?

Mar 1, 2010

g_edgar

First you will need a definition of [tex]e^x[/tex] ... what is yours?

1. Why is e^x equal to (1+x/n)^n as n approaches Infinity?

This is because as n approaches Infinity, the term (1+x/n)^n approaches the definition of the mathematical constant e. This can be seen through the limit definition of e, which states that as n approaches Infinity, (1+1/n)^n approaches the value of e.

2. Can you prove why e^x is equal to (1+x/n)^n as n approaches Infinity?

Yes, this can be proven using the limit definition of e and the properties of limits. By taking the limit as n approaches Infinity of (1+x/n)^n, we can see that it approaches the definition of e. Therefore, e^x and (1+x/n)^n are equivalent as n approaches Infinity.

3. How does the value of e relate to the equation e^x = (1+x/n)^n as n approaches Infinity?

The equation e^x = (1+x/n)^n as n approaches Infinity is a representation of the definition of e. As n approaches Infinity, the term (1+x/n)^n approaches the value of e, making the equation true.

4. Is the equation e^x = (1+x/n)^n as n approaches Infinity only valid for certain values of x?

No, the equation is valid for all values of x. This can be seen through the limit definition of e, which does not have any restrictions on the values of x.

5. Can you provide an example of how the equation e^x = (1+x/n)^n as n approaches Infinity is used in real life?

One example of how this equation is used in real life is in compound interest calculations. The equation e^x = (1+x/n)^n as n approaches Infinity can be used to model the growth of an investment over time, where x represents the annual interest rate and n represents the number of compounding periods per year.