+- HP Forums (https://www.hpmuseum.org/forum)
+-- Forum: HP Calculators (and very old HP Computers) (/forum-3.html)
+--- Forum: HP Prime (/forum-5.html)
+--- Thread: XCas sum limit confusion (/thread-15503.html)
XCas sum limit confusion - Albert Chan - 08-26-2020 07:48 PM
Quote:Discrete sum (with 2 or 4 arguments return then sum from a to b if a<=bAbove quote is from XCas help screen for sum()
or of the opposite of the sum from b+1 to a-1 if a>b+1 or 0 if a=b+1)
or the discrete primitive or sum of the elements of a list or a sequence.
sum(Expr,Var,VarMin(a),VarMax(b),[VarStep(p)])
Why the convoluted swapping of limit with aboved logic ?
And, why swapped limit changed to b+1 to a-1, and not b to a ?
Some confusing examples:
XCas> sum(k, k, 1, 5) // = 1+2+3+4+5 = 15
XCas> sum(k, k, 5, 1) // = -sum(k, k, 2, 4) = -(2+3+4) = -9. OK, but don't know why
XCas> sum(k, k, 5, 1, 1) // got 15 ? somehow, step flipped to -1, instead of swapping limits.
XCas> sum(k, k, 5.1, 1) // got -14.0 ? expected -sum(k, k, 2, 4.1) = -(2+3+4) = -9
Missing step is not equivalent to default step=1
XCas> factor(sum(k, k, a, b)) // (b+a)*(b-a+1)/2
XCas> sum(k, k, a, b, 1) // "Unable to sort boundaries a,b Error: Bad Argument Value"
RE: XCas sum limit confusion - Albert Chan - 08-26-2020 09:58 PM
This may explain the reason for (a,b) → (b+1,a-1), if a>b+1
Say, we have a function F(x) = sum(f(t), t=-inf .. x-1)
If a ≤ b, S1 = sum(f(t), t=a .. b) = F(b+1) - F(a)
If a > b, we could flip the limit, just like doing integrals.
S2 = - sum(f(t), t=b .. a) = -(F(a+1) - F(b)) = F(b) - F(a+1)
But, S1 != -S2. To aim for symmetry, we shift the limit a bit:
S2 = - sum(f(t), t=b+1 .. a-1) = F(a) - F(b+1) = -S1
Sympy Gamma: Sum(k, (k, 5, 1)) = -(2 + 3 + 4) = -9
But, the cure maybe worse than the disease.
Mathematica also does closed end limit, but generated a list. (conceptually)
If the list is empty, there is nothing to sum.
We lost the symmetry, but it is simple to understand.
Mathematica: Sum(k, (k, 5, 1)) = 0
Open-ended sum is very elegant:
S1 = sum(f(t), t = a .. b-1) = F(b) - F(a)
S2 = sum(f(t), t = b .. a-1) = F(a) - F(b)
We got nice symmetry, without worrying a, b sort order.
Also of interest: Should array indices start at 0 or 1 ?