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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