April 8, 2010

Sum[i,{i,1,n}]

Sum[i^2,{i,1,n}]

Sum[i^3,{i,1,n}]

Sum[i^4,{i,1,n}]

Sum[3i^13+7i^5+4i+5,{i,1,n}]

% /.n->13

Sum[3i^13+7i^5+4i+5,{i,1,13}]

f[n_] := N[ Sum[ 1/i!,{i,0,n}],30]

f[5]

f[100]

f[1000]

g1[n_] := N[ Sum[ (-1)^i*(Pi/2)^(2i)/(2i)!, {i,0,n}],25]

g1[5]

g2[n_] := N[ Sum[ (-1)^i*(Pi/3)^(2i)/(2i)!, {i,0,n}],25]

g2[5]

g2[25]

g3[n_] := N[ Sum[ (-1)^i*(Pi/4)^(2i)/(2i)!, {i,0,n}],25]

g3[25]

N[Sqrt[2]/2,25]

g4[n_] := N[ Sum[ (-1)^i*(Pi/6)^(2i)/(2i)!, {i,0,n}],25]

g4[25]

N[Sqrt[3]/2,25]

h1[n_]:= N[ Sum[ (-1)^i*(Pi/2)^(2i+1)/(2i+1)!,{i,0,n}],25]

h1[5]

h1[25]

h2[n_]:= N[ Sum[ (-1)^i*(Pi/3)^(2i+1)/(2i+1)!,{i,0,n}],25]

h2[115]

h3[n_] :=  N[ Sum[ (-1)^i*(Pi/4)^(2i+1)/(2i+1)!,{i,0,n}],25]

h3[25]

h4[n_] :=  N[ Sum[ (-1)^i*(Pi/6)^(2i+1)/(2i+1)!,{i,0,n}],25]

h4[25]

Sin[Pi/6] //N

N[Sqrt[2]/2,25]

N[Sqrt[3]/2,25]

There is a constant C (the Euler-Mascheroni constant)  such that 
            (Underscript[Limit, n->∞]) ( (Underoverscript[ ∑, (i=1), n])(1/i)   - Log(n) ) = C.     (Here, Log is the natural logarithm.)
Find an approximate numerical value for C.

H[n_] := N[Sum[1/i,{i,1,n}] - Log[n], 10]

H[5]

Table[{n,H[n]},{n,100,10000,100}]

H[100000]

Evaluate  (Underoverscript[((π^2)/6)- ∑, (                i=1), (               n)])1/i^2    for n = 5, 100, 1000, 10000, 100000

F[n_] :=N[   Pi^2/6 -  Sum[ 1/i^2, {i,1,n}], 20]

F[5]

F[100]

F[1000]

F[10000]

F[100000]

Evaluate   ((π^4)/90) - (∑_(i=1)^n)(1/(i^4)),  for   n = 5, 25, 100

G[n_] := N[ Pi^4/90 - Sum[1/i^4,{i,1,n}], 20]

G[5]

G[25]

G[100]

G[1000]

Evaluate   ((π^6)/945) - (∑_(i=1)^n)(1/(i^6)),  for   n = 5, 25, 100

H[n_] := N[ Pi^6/945 - Sum[1/i^6,{i,1,n}], 20]

H[5]

H[25]

H[100]

H[1000]

(Underoverscript[Find an integer k so that the formula   ∑, (                                                                                                       i=1), (                                                                                                      ∞)])(1/(i^8)) = ((π^8)/k)  appears to be true.

Quit[]

Clear[k]

f[n_] := Pi^8/k - Sum[1/i^8,{i,1,n}]

f[25]

Solve[ f[25]==0,k]//N

g[n_] := N[ Pi^8/9450 - Sum[1/i^8,{i,1,n}], 25]

g[100]

g[n_] := Pi^10/K - Sum[1/i^10,{i,1,n}]

g[25]

Solve[g[25]==0,K] // N

h[n_] := N[ Pi^10/93555-Sum[1/i^10,{i,1,n}],25]

h[50]