6
seq = Table[Sum[(1 + Cos[k Pi/n])^n, {k, 1, n - 1}], {n, 10}] //
FullSimplify;
sum1 = FindSequenceFunction[seq, n] // FullSimplify
(* 2^(-1 + n) (-1 + (2 Gamma[1/2 + n])/(Sqrt[π] Gamma[n])) *)
To convert the ratio of Gamma functions to a Binomial function
repl = Gamma[a_]/Gamma[b_] :>
Gamma[1 + a - b] Binomial[a - 1, b - 1];
sum2 = sum1 /. ...
5
Clear[f]
Use Piecewise
f[s_] := Sum[
Piecewise[{{MoebiusMu[n]/n Log[Zeta[n s] (n s - 1)],
n != 1/s}}],
{n, 1, 10}];
f[1]
(* -(1/2) Log[π^2/6] + 1/6 Log[π^6/189] +
1/10 Log[π^10/10395] - 1/3 Log[2 Zeta[3]] - 1/5 Log[4 Zeta[5]] -
1/7 Log[6 Zeta[7]] *)
% // N
(* -0.591981 *)
f'[1]
% // N
(* -0.850562 *)
Plot[{f'[s], f[s]}, {s, 0, 10},
...
4
You can use ReplaceAll:
sum = Sum[Subscript[b, k] x^k, {k, 0, Infinity}];
x sum /. a_. Sum[s_, r__] :> Sum[a s, r]
TeXForm @ %
$\sum _{k=0}^{\infty } b_k x^{k+1}$
3
First, I explored the meaning of the error message:
NSum[f[n], {n, 1, 100}]
NSum::nsnum: Summand (or its derivative) [Piecewise] <<1>> is not numerical at point n = 16.
The error message suggests evaluating f and its derivative at 16:
f[16]
f'[16]
(*
0.0454989
-0.0802852 + 0.0523825 Derivative[1][Re][16]
*)
Aha! The derivative of f has a ...
2
I tried:
decompose[p1_, p2_] :=
Module[{x1 = p1, x2 = p2},
If[x1 <= x2, count = 0;
Do[count++;
Print[k1, x2 - x1 + 2 k2, k1], {k1, 0, x1}, {k2, 0, x1 - k1}];
Print["Total of ", count],
Print["Wrong values for the p1 and the p2"]]];
decompose[2, 2];
The output is:
000
020
040
101
121
202
Total of 6
The do loop function ...
1
Define a function that can only be called with a numerical argument $i$:
f[i_?NumericQ] := NIntegrate[i + x, {x, 1, 7}]
Numerical sum without possibility of analytic attempts at integration:
NSum[f[i], {i, 1, 7}]
(* 336. *)
1
You can use DistanceMatrix and use Total with the upper part of the matrix:
rowtotaldistances = Total[UpperTriangularize[DistanceMatrix[#],1],{2}]&
m = Partition[Range[0,10],2];
TeXForm @ MatrixForm @ m
$\left(
\begin{array}{cc}
0 & 1 \\
2 & 3 \\
4 & 5 \\
6 & 7 \\
8 & 9 \\
\end{array}
\right)$
TeXForm @ MatrixForm[...
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