where is the usual binomial coefficient.
Leibniz rule (generalized product rule - Wikipedia, the free encyclopedia)
anyone? its not given at wikipedia.
where is the usual binomial coefficient.
Leibniz rule (generalized product rule - Wikipedia, the free encyclopedia)
anyone? its not given at wikipedia.
I am considering both functions f and g to be functions of the single variable x.
Consider the case for n = 1:
So the theorem holds for n = 1.
Now assume it holds for some n = N, that is:
We wish to show it holds for n = N + 1 also.
So:
We need to show that this is equal to
There is going to be a way to do this in general, but you should probably start small. Let N = 1, for example. Then we are saying:
The advantage is that we can write this out term by term and see what is happening:
See if you can you use this example and maybe the next few higher values of N to find a way to set up a conditional equation between the two summations.
I'll give you a hint: It's the same pattern as you see when you are proving that
-Dan
Thanks for the hint.
I'm guessing there is a need to utilise the Binomial Theorem:
1) I'm wondering how should I get
reduced to .
My only solution so far is to extract the upperbound term from the summation.
2) How to deal with the binomial coefficient? Should I use the identity
They don't lead anywhere as of now... but I'll keep trying.
Ok, so far what I've figured out is to do it by brute force:
Then, using the identity (which is easily proven):
the equation can be re-arranged into:
which is actually the expanded form of
I hope there is a better way to do it though. This one doesn't seem elegant enough.