Monday, April 18, 2016

How to convert a series to an equation?










I don't know the technical language for what I'm asking, so the title might be a little misleading, but hopefully I can convey my purpose to you just as well without.



Essentially I'm thinking of this: the series 4n+4n14nn.




I suppose this is the summation of the series 4n from n to 0.



But is there any way to express this as a pure equation, not as a summation of a series?



If so, how do you figure out how to convert it?


Answer



In general, for x1 it is true that
nk=0xk=1+x++xn=xn+11x1.


So, in your case in particular, we have that

nk=04nk=4n++4+1=1+4++4n=nk=04k=4n+113.

Alternatively, one could pull out a factor of 4n from all terms, and compute
nk=04nk=4nnk=0(14)k=4n(14)n+11(14)1=4n4n+114n+134=4n+14n+114n+13=4n+113.


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