Posted by T.Gracken on September 12, 2002 at 13:42:44:
In Reply to: Very confusing... posted by Joel on September 12, 2002 at 09:57:08:
: : : : : Is a Sum statement properly termed a "function"
: : : : : so that EITHER of the following:
: : : : : f(n) = n(n+1)/2
: : : : : and
: : : : : f(n) = Sum[from n=1 to n]n
: : : : : BY ITSELF would be correctly considered a function that defines the series {1,3,6,10,15...} ?
: : : : : The former, of course, is a quadratic function. What is the latter? Is it a propositional function?
: : : : You wrote the latter
: : : : f(n) = Sum[from n=1 to n]n
: : : : wrong
: : : : It should be
: : : : f(n) = Sum[from j=1 to n]j
: : : : Then f(n)= n(n+1)/2
: : : : Yes f(n)=n(n+1)/2 defines the series
: : : : {1,3,6,10,15...}
: : : :
: : : Hmmm... now that you pointed it out I agree that
: : : f(n) = Sum[from n=1 to n]n
: : : seems to have too many n's.
: : : But I don't think your f(n) = Sum[from j=1 to n]j looks right either.
: : :
: : : Shouldn't it be:
: : : f(j) = Sum[from j=1 to n]j ?
: So, is this a more general (and correct) way to express this type of relationship:
: f(n) = Sum[from j=1 to n](f(j))
no
just write: f(n) = sum[from j=1 to n] j
...so f(1)=1, f(2)=1+2, f(3)=1+2+3, etc.
It is not incorrect to have f(n) = Sum[from j=1 to n](g(j)), but you would need to know what g(j) equals in order to evaluate f(n).{i.e. f(3)=g(1)+g(2)+g(3)}
you could define a recursive function using notation similar to f(n)=sum[from j=1 to n](f(j)), but there is definitely a problem here.
note, for what you wrote; f(1) = f(1), f(2) = f(1)+f(2), f(3) = f(1)+f(2)+f(3), etc.
recall that sigma notation is simply a shorthand way for writing a sum of a sequence of terms.
...where a sequence is a list of numbers, and a series is the sum of the numbers in the list.
: and in the specific case we were discussing, f(j) happened to be simply j?