How to Simplify Factorials with a Proof for k*(k!)=(k+1)!-1?

In summary, the conversation discusses an attempt to prove k * (k!) = (k + 1)! - 1, which is found to be false for all values of k except k = 1. The conversation also considers the possibility of the question being k * k! = (k + 1)! - k!, but this is also found to be false. The final solution is to use the equation 1*1!+2*2!+...+k*k!=(k + 1)! - 1 to prove the original statement.
  • #1
kathrynag
598
0

Homework Statement



I'm trying to prove k*(k!)=(k+1)!-1

Homework Equations





The Attempt at a Solution


This is how far I've gotten:
k[k(k-1)(k-2)...1)]
 
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  • #2
What you are trying to prove is not true for all k.
 
  • #3
Ok, but then what would I do since I know it's true for k=1
 
  • #4
Are you sure the question doesn't say k * k! = (k + 1)! - k!?
 
  • #5
No, I'm positive. Just checked in the book.
 
  • #6
Plug in k = 2 to see that it's false.
 
  • #7
Ok, so if I was given this question. I just write only true for k=1?
 
  • #8
mutton said:
Are you sure the question doesn't say k * k! = (k + 1)! - k!?

Well it's 1*1!+2*2!+...+k*k!=(k + 1)! - k!
 
  • #9
kathrynag said:
Well it's 1*1!+2*2!+...+k*k!=(k + 1)! - k!

That's not true.

[tex]k * k! = (k + 1 - 1) k! = (k + 1) k! - 1 * k! = (k + 1)! - k![/tex]
 
  • #10
kathrynag said:
Well it's 1*1!+2*2!+...+k*k!=(k + 1)! - k!

1*1!+2*2!+...+k*k!=(k + 1)! - 1
sorry...
 
  • #11
kathrynag said:
1*1!+2*2!+...+k*k!=(k + 1)! - 1
sorry...

Okay, so how do you plan to prove this?
 
  • #12
Never mind. Just figured it out!
 

1. What is factorial simplifying?

Factorial simplifying is a mathematical process of simplifying expressions containing factorial notation. Factorial notation is denoted by an exclamation mark (!) and represents the product of all positive integers less than or equal to the given number.

2. How is factorial simplifying useful?

Factorial simplifying is useful in many areas of mathematics, such as combinatorics, probability, and calculus. It allows for simpler and more efficient calculations and can help identify patterns in complex expressions.

3. What are the basic rules of factorial simplifying?

The basic rules of factorial simplifying include the product rule (n! * m! = (n+m)!), the quotient rule (n! / m! = (n-m)!), and the power rule ((n!)^m = n! * (n-1)! * (n-2)! * ... * (n-m+1)!).

4. Can factorial simplifying be applied to any expression?

No, factorial simplifying can only be applied to expressions containing factorial notation. It is not applicable to other types of mathematical notation, such as exponential or logarithmic notation.

5. Are there any common mistakes to avoid when simplifying factorials?

Yes, some common mistakes to avoid when simplifying factorials include incorrectly expanding or canceling out terms, forgetting to include the necessary factors in the final answer, and using the wrong rules for simplifying. It is important to double-check all steps and follow the rules carefully to avoid errors.

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