Wolfram Alpha != HP Prime CAS result: who is wrong?

[Albert Chan]

HP Prime behavior Wrote:\( \displaystyle \sum_{k=0}^{-n} k = \frac12(-n)(-n+1) = \frac12n(n-1) = \frac12(n-1)(n-1+1) = \sum_{k=0}^{n-1} k \)

It's not deep, a bit entertaining perhaps.

With limit = 0 .. n-1, many sum formulas are simpler.
Off by 1 is a result of forward difference operator.

Δf(x) = f(x+1) - f(x)
Σ(Δf(x), x=0 .. n-1) = f(x+n) - f(x)

Example, in falling factorial form, sum is just like integration.

\( \displaystyle \sum_{k=0}^{n-1} k^{\underline n} =
\left.
\frac{k^{\underline {n+1}}}{n+1}
\right| _0 ^n
\)

Perhaps this is why HP Prime internally work with open-ended sum formula.
Note: sign of n does not matter.

\( \displaystyle \sum_{k=0}^{-n} k
= \left. \frac{k^{\underline {2}}}{2} \right| _0 ^{-n+1}
= \left. \frac{k^{\underline {2}}}{2} \right| _0 ^{n}
= \sum_{k=0}^{n-1} k
\)