6
sumRule =
Sum[expr1_, iter_List] + Sum[expr2_, iter_List] :>
Sum[expr1 + expr2, iter];
productRule =
Product[expr1_, iter_List]*Product[expr2_, iter_List] :>
Product[expr1*expr2, iter];
Sum[Subscript[x, i], {i, 1, n}] +
Sum[-Subscript[x, i], {i, 1, n}] /. sumRule
(* 0 *)
Product[Subscript[x, i], {i, 1, n}]*
Product[Subscript[x, i]...
4
rule1 = Sum[Times[a_., b_], c_] :> a Sum[b, c];
rule2 = Product[Power[a_, b_.], c_] :> Product[a, c]^b;
Examples:
expr1 = Sum[Subscript[x, i], {i, 1, n}] + Sum[-Subscript[x, i], {i, 1, n}];
TeXForm @ expr1
$\sum _{i=1}^n -x_i+\sum _{i=1}^n x_i$
expr1 /. rule1
0
expr2 = Product[Subscript[x, i], {i, 1, n}]*Product[Subscript[x, i]^-1, {i, 1, n}]...
3
Let me show you an example of series solution of ODE with given f0 and f1 for your differential equation.
{f0[z_] = Sin[z], f1[z_] = 1};
deq = Derivative[2][w][z] + f1[z]*Derivative[1][w][z] + f0[z] == 0
For comparison the solution with DSolve at initial conditions.
wdsol = w /. First@DSolve[deq && w[0] == 4 && w'[0] == 3, w, z]
(* ...
2
Mathematica 12.0 also claims the divergence of s2. I think the defect under consideration is not hard in view of a workaround
Sum[HarmonicNumber[n] - (EulerGamma + Log[n] + 1/(2 n) - 1/(12 n^2)), {n, 1, a},
Assumptions -> a > 1]
1/12 (-12-12 a-12 a EulerGamma-6 HarmonicNumber[a]+12 HarmonicNumber[1+a]+12 a HarmonicNumber[1+a]+HarmonicNumber[a,2]-...
2
If you apply FunctionExpand to the expression with an explicit value of k, but with n undefined, then you should be able to get rid of the 0/0 issues:
sum[n_, k_] := Block[{m},
FunctionExpand[Binomial[m,k] Hypergeometric2F1[-k,-3-m,1-k+m,2]] /. m->n
]
Examples:
sum[1,2]
sum[1,5]
sum[1,6]
sum[2,7]
sum[2,8]
32
16
0
32
0
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