Sum
Symbolic summation.
$ wo 'Sum[k, {k, 1, 10}]'
55
$ wo 'Sum[k^2, {k, 1, n}]'
(n*(1 + n)*(1 + 2*n))/6
$ wo 'Sum[1/k^2, {k, 1, Infinity}]'
Pi^2/6
$ wo 'Sum[(-1)^n x^(2n+1)/Factorial[2n+1], {n, 0, Infinity}]'
Sin[x]
A trailing option is not an iterator, so it leaves the sum alone:
$ wo 'Sum[n, {n, 1, 10}, Method -> Automatic]'
55
Regularization assigns a value to a divergent sum. "Dirichlet" sums n^k as
the Dirichlet series it continues, giving Zeta[-k]:
$ wo 'Sum[n, {n, 1, Infinity}, Regularization -> "Dirichlet"]'
-1/12
$ wo 'Sum[n^3, {n, 1, Infinity}, Regularization -> "Dirichlet"]'
1/120
"Abel" applies to an alternating summand:
$ wo 'Sum[(-1)^n, {n, 1, Infinity}, Regularization -> "Abel"]'
-1/2
$ wo 'Sum[(-1)^n n, {n, 1, Infinity}, Regularization -> "Abel"]'
-1/4
A summand neither scheme reaches keeps the call — 1/n lands on the pole of
Zeta at 1:
$ wo 'Sum[1/n, {n, 1, Infinity}, Regularization -> "Dirichlet"]'
Sum[n^(-1), {n, 1, Infinity}, Regularization -> Dirichlet]