Proof by Induction: Understanding D/dx Formula

In summary: Thanks for your help!In summary, the author is showing that if n=k, then n=k+1 is automatically true. They are trying to show that n=k implies n=k+1, but they are not explain how to get from (-1)^k to (-1)^k+1.
  • #1
sony
104
0
Ok so I have found a formula for d^n/dx^n 1/x^2
= (-1)^n * (1+n)! * x^-(n+2)

So I have to do d/dx [(-1)^n * (1+n)! * x^-(n+2)] and see what I end up with. But how do I do that.

My book gives an example: (from d/dx (1+x)^-1)
d/dx [(-1)^k * k!(1+x)^(-k-1)] = (-1)^k * k!(-k-1)(1+x)^(-k-2)=...

What on Earth is going on?! My book just drops explaning _how_ . Where does (-k-1) come from? I'm stuck...

Thanks
 
Physics news on Phys.org
  • #2
sony said:
Ok so I have found a formula for d^n/dx^n 1/x^2
= (-1)^n * (1+n)! * x^-(n+2)

So I have to do d/dx [(-1)^n * (1+n)! * x^-(n+2)] and see what I end up with. But how do I do that.

My book gives an example: (from d/dx (1+x)^-1)
d/dx [(-1)^k * k!(1+x)^(-k-1)] = (-1)^k * k!(-k-1)(1+x)^(-k-2)=...

What on Earth is going on?! My book just drops explaning _how_ . Where does (-k-1) come from? I'm stuck...

Thanks


the essence of proof by induction is to show that

a) there is a minimum case where what you want to prove is true
b) that if n=k is true, it follows automatically (after some manipulation) that n=k+1 is true.

there is a theorem that says that if these conditions are satisfied, the statement in question is true.


so the authors are trying to show that n=k implies that n=k+1.
 
  • #3
Brad Barker said:
the essence of proof by induction is to show that

a) there is a minimum case where what you want to prove is true
b) that if n=k is true, it follows automatically (after some manipulation) that n=k+1 is true.

there is a theorem that says that if these conditions are satisfied, the statement in question is true.


so the authors are trying to show that n=k implies that n=k+1.
Yes I get _that_ :)

But how do I do the derivative of that expression?
 
  • #4
sony said:
Ok so I have found a formula for d^n/dx^n 1/x^2
= (-1)^n * (1+n)! * x^-(n+2)

So I have to do d/dx [(-1)^n * (1+n)! * x^-(n+2)] and see what I end up with. But how do I do that.

Reread the chapter on derivatives. (d/dx)(x^n) = n*x^(n-1).
 
  • #5
Nevermind, it was how to get from (-1)^k to (-1)^k+1 and the faculty thing I didnt get.
 

1. What is proof by induction?

Proof by induction is a mathematical technique used to prove that a statement holds for all natural numbers. It involves two steps: a base case where the statement is shown to be true for the first natural number, and an inductive step where the statement is shown to be true for the next natural number assuming it is true for the previous one.

2. How does proof by induction work?

In proof by induction, the base case is first established by directly showing that the statement is true for the first natural number (usually 1). Then, the inductive step involves showing that if the statement is true for some natural number k, it is also true for the next natural number k+1. This process can then be repeated for any subsequent natural number, proving that the statement holds for all natural numbers.

3. What is the D/dx formula used for?

The D/dx formula, also known as the derivative formula, is used to calculate the derivative of a function at a specific point. It tells us how much the function changes with respect to the independent variable at that point.

4. How is the D/dx formula used in proof by induction?

In proof by induction, the D/dx formula is used to calculate the derivative of the statement being proven at each step, starting with the base case and using the inductive hypothesis. If the derivative is shown to be true for the first natural number, and the inductive step proves that it is also true for the next natural number, then it can be concluded that the statement holds for all natural numbers.

5. Can proof by induction be used to prove any statement?

No, proof by induction can only be used to prove statements that are true for all natural numbers. It cannot be used for statements that involve real numbers or other types of numbers. It is also important to note that the statement being proven must be well-defined and precise for proof by induction to be valid.

Similar threads

  • Introductory Physics Homework Help
Replies
25
Views
1K
  • Introductory Physics Homework Help
Replies
14
Views
2K
  • Introductory Physics Homework Help
Replies
1
Views
1K
  • Introductory Physics Homework Help
Replies
1
Views
752
  • Introductory Physics Homework Help
Replies
2
Views
1K
  • Introductory Physics Homework Help
Replies
22
Views
444
  • Introductory Physics Homework Help
Replies
1
Views
709
  • Introductory Physics Homework Help
Replies
2
Views
484
  • Introductory Physics Homework Help
Replies
28
Views
268
  • Introductory Physics Homework Help
Replies
3
Views
802
Back
Top