Is a variable that depends on ? What is the relationship?[/tex]
Prove that
where is a nonnegative integer and is any real number, which is not a negative integer.
[Hint: Use the method of mathematical induction and integration by parts.]
I do not understand how to proceed with induction?
Ok, I get it now. The dt was throwing me. With dx, it actually seems fairly trivial, so I may be missing something.
Base case: n = 0
So, it satisfies the base case. Suppose it is true for all nonnegative integers less than . I want to show it is true for .
The integration by parts is very straightforward: .
Now, we have:
The first term is zero at both 1 and 0. For the second term, since and is a nonnegative integer less than , so by the induction assumption, we can apply the hypothesis.
So based on your help, this is my proof. Does this seem accurate?
Problem:
where is a nonnegative integer and is any real number, which is not a negative integer.
[Hint: Use the method of mathematical induction and integration by parts.]
Proof: We will prove this by Mathematical Induction.
We need to show that it holds for n=0. Now,
and this simplifies to:
.
Hence it holds for n=0. Now, suppose that it is true for , we need to show that it holds for .
First, evaluate the integral using integration by parts. Let . So now, we have:
.
Now, = 0 + 0 = 0.
Now by our assumption we have , so now consider .
.
Where do I go about applying the hypothesis of n = k?