Sum

Sum(expr, {i, imin, imax})

evaluates the discrete sum of expr with i ranging from imin to imax.

Sum(expr, {i, imax})

same as Sum(expr, {i, 1, imax}).

Sum(expr, {i, imin, imax, di})

i ranges from imin to imax in steps of di.

Sum(expr, {i, imin, imax}, {j, jmin, jmax}, ...)

evaluates expr as a multiple sum, with {i, ...}, {j, ...}, ... being in outermost-to-innermost order.

See

• Wikipedia - Summation

Examples

>> Sum(k, {k, 1, 10})
55

Double sum:

>> Sum(i * j, {i, 1, 10}, {j, 1, 10})
3025

>> Table(Sum(i * j, {i, 0, n}, {j, 0, n}), {n, 0, 4})
{0,1,9,36,100}

Symbolic sums are evaluated:

>> Sum(k, {k, 1, n})
1/2*n*(1+n)

>> Sum(k, {k, n, 2*n})
3/2*n*(1+n)

>> Sum(k, {k, I, I + 1})
1+I*2

>> Sum(1 / k ^ 2, {k, 1, n})
HarmonicNumber(n, 2)

Verify algebraic identities:

>> Simplify(Sum(x ^ 2, {x, 1, y}) - y * (y + 1) * (2 * y + 1) / 6)
0

Infinite sums:

>> Sum(1 / 2 ^ i, {i, 1, Infinity})
1

>> Sum(1 / k ^ 2, {k, 1, Infinity})
Pi^2/6

>> Sum(x^k*Sum(y^l,{l,0,4}),{k,0,4})
1+y+y^2+y^3+y^4+x*(1+y+y^2+y^3+y^4)+(1+y+y^2+y^3+y^4)*x^2+(1+y+y^2+y^3+y^4)*x^3+(1+y+y^2+y^3+y^4)*x^4

>> Sum(2^(-i), {i, 1, Infinity})
1

>> Sum(i / Log(i), {i, 1, Infinity})
Sum(i/Log(i),{i,1,Infinity})

>> Sum(Cos(Pi i), {i, 1, Infinity})
Sum(Cos(i*Pi),{i,1,Infinity})

Implementation status

• ☑ - partially implemented

Github

• Implementation of Sum

• Rule definitions of Sum