Proving that the sum of (u^x)/x from 0 to infinity = e^u

In summary, the conversation discusses the derivation of the formula for the sum of (u^x)/x! from 0 to infinity, which is equal to e^u. One participant suggests checking the MacLaurin series expansion for e^x, while another confirms that this approach is sufficient for a proof.
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  • #2
Hi CraigH! :smile:

Isn't that the definition of eu ?

(and you can check it by working out euev, or by differentiating it :wink:)
 
  • #3
Do you know about the MacLaurin series expansion ? What is it for ex ?
 
  • #4
Ah yes, I've just looked up the MacLaurin series for e^x in my data book, its give me the answer i was hoping for. I needed to know as part of a proof for a different equation.

So in my exam just saying:

"From the MacLaurin series:
The sum of (u^x)/x! from 0 to infinity = e^u"

Should be enough then.

Thank you for answering.
 

1. What is the concept behind proving that the sum of (u^x)/x from 0 to infinity equals e^u?

The concept behind this proof is related to the mathematical concept of convergence, which is the idea that a series of values approaches a specific limit as the number of terms in the series increases. In this case, we are trying to show that as the number of terms in the series (u^x)/x increases, the sum approaches the value of e^u.

2. How is the proof for this equation typically approached?

The proof for this equation is typically approached using techniques from calculus and real analysis, such as the limit definition of a sum and the definition of the exponential function. It involves manipulating the terms of the series and taking limits to show that the sum converges to e^u.

3. What is the significance of this equation in mathematics?

This equation is significant because it shows the relationship between the exponential function and the concept of convergence. It also has practical applications in fields such as physics, engineering, and economics, where exponential growth and decay are often observed.

4. Can this equation be generalized to other values of x and u?

Yes, this equation can be generalized to other values of x and u. In fact, it is often used as a starting point for proving more general identities and theorems related to convergence and the exponential function.

5. What are some real-world examples of this equation in action?

One example of this equation in action is in compound interest calculations, where the sum of (u^x)/x represents the total amount of money earned over time with a continuously compounded interest rate of e^u. It is also used in population growth models, electrical circuit analysis, and radioactive decay calculations.

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