We know that if 
then the sum converges to
Coming from the other side and using binomial theorem we get
which expands to
Which leads to an interesting conclusion
I got maxima to verify this by calling the binomial function for various combinations of -1 and n. I am a little confused about the interpretation of the meaning though
Usually nC2 is used to imply from "n" objects, choose 2 at a time. nC2 gives us the number of total such combinations. What does -1Cn mean? What are the other proofs for -1Cn?
Usually nC2 is used to imply from "n" objects, choose 2 at a time. nC2 gives us the number of total such combinations. What does -1Cn mean? What are the other proofs for -1Cn?
2 comments:
Binomial Theorem applies only to non-negative integers for the exponent.
http://en.wikipedia.org/wiki/Binomial_theorem
( -1 choose n doesn't make any sense )...
You can get the expansions here ( Wolfram Alpha )
http://www.wolframalpha.com/input/?i=1%2F(1-x)
Agreed, -1 makes no sense by definition, but we need to ask ourselves why? In my case it is probably just coincidental that the result I got agreed with the output.
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