Tässä yksi tapa eli minun tapani ratkaista kuudennet mikroharjoitukset.
Markko
Input := <<Algebra`SymbolicSum`
Input := Sum[1/((k-1)(k+1)),{k,2,Infinity}]
Output = 3 - 4
Input := Sum[1/((k-1)(2k+1)),{k,2,Infinity}]
Output = 2 (4 - 3 Log[2]) ---------------- 9
Input := Sum[1/((k^2-1)(k^2+1)),{k,2,Infinity}]
Output = 7 Pi Coth[Pi] - - ----------- 8 4
Input := Sum[1/k^1,{k,1,Infinity}]
Output = Infinity
Input := Sum[1/k^2,{k,1,Infinity}]
Output = 2 Pi --- 6
Input := Sum[1/k^3,{k,1,Infinity}]
Output = Zeta[3]
Input := Sum[1/k^4,{k,1,Infinity}]
Output = 4 Pi --- 90
Input := Sum[1/k^alfa,{k,1,Infinity}]
Output = Zeta[alfa]
Input := Zeta[0]
Output = 1 -(-) 2
Input := Zeta[1]
Output = ComplexInfinity
Input := ?Series
Series[f, {x, x0, n}] generates a power series expansion for f about the point x = x0 to order (x - x0)^n. Series[f, {x, x0, nx}, {y, y0, ny}] successively finds series expansions with respect to y, then x.
Input := Series[ArcTan[x],{x,0,5}]
Output = 3 5 x x 6 x - -- + -- + O[x] 3 5
Input := Series[ArcTan[x],{x,0,10}]
Output = 3 5 7 9 x x x x 11 x - -- + -- - -- + -- + O[x] 3 5 7 9
Input := Series[ArcTan[x],{x,0,20}]
Output = 3 5 7 9 11 13 15 17 19 x x x x x x x x x 21 x - -- + -- - -- + -- - --- + --- - --- + --- - --- + O[x] 3 5 7 9 11 13 15 17 19
Input := Plot[Evaluate[{ArcTan[x],Normal[%19],Normal[%20],Normal[%21]}] ,{x,-5,5},PlotRange->{-5,5}]
Output = -Graphics-
Input := SymbolicSum[x^k/k,{k,1,Infinity}]
Output = -Log[1 - x]
Input := Plot[{Sum[x^k/k,{k,1,10}],-Log[1-x]},{x,-5,5},PlotRange->{-7,7}]
Plot::plnr: CompiledFunction[{x}, <<1>>, -CompiledCode-][x] is not a machine-size real number at x = 1.25.
Plot::plnr: CompiledFunction[{x}, <<1>>, -CompiledCode-][x] is not a machine-size real number at x = 1.04167.
Plot::plnr: CompiledFunction[{x}, <<1>>, -CompiledCode-][x] is not a machine-size real number at x = 1.01563.
General::stop: Further output of Plot::plnr will be suppressed during this calculation.
Output = -Graphics-
Input := SumbolicSum[Cos[k^2x]/k^2,{k,1,Infinity}]
Output = 2 Cos[k x] SumbolicSum[---------, {k, 1, Infinity}] 2 k
Input := Plot[Sum[Cos[k^2x]/k^2,{k,1,20}],{x,-5,5},PlotRange->{-2,2}]
Output = -Graphics-
Input := Series[Cos[x],{x,0,10}] * (Sum[a[k] x^k, {k,0,10}] + O[x]^11 )
Output = -a[0] 2 -a[1] 3 a[0] + a[1] x + (----- + a[2]) x + (----- + a[3]) x + 2 2 a[0] a[2] 4 a[1] a[3] 5 (---- - ---- + a[4]) x + (---- - ---- + a[5]) x + 24 2 24 2 -a[0] a[2] a[4] 6 (----- + ---- - ---- + a[6]) x + 720 24 2 -a[1] a[3] a[5] 7 (----- + ---- - ---- + a[7]) x + 720 24 2 a[0] a[2] a[4] a[6] 8 (----- - ---- + ---- - ---- + a[8]) x + 40320 720 24 2 a[1] a[3] a[5] a[7] 9 (----- - ---- + ---- - ---- + a[9]) x + 40320 720 24 2 -a[0] a[2] a[4] a[6] a[8] 10 11 (------- + ----- - ---- + ---- - ---- + a[10]) x + O[x] 3628800 40320 720 24 2
Input := LogicalExpand[%39==1]
Output = -a[0] -1 + a[0] == 0 && a[1] == 0 && ----- + a[2] == 0 && 2 -a[1] a[0] a[2] ----- + a[3] == 0 && ---- - ---- + a[4] == 0 && 2 24 2 a[1] a[3] ---- - ---- + a[5] == 0 && 24 2 -a[0] a[2] a[4] ----- + ---- - ---- + a[6] == 0 && 720 24 2 -a[1] a[3] a[5] ----- + ---- - ---- + a[7] == 0 && 720 24 2 a[0] a[2] a[4] a[6] ----- - ---- + ---- - ---- + a[8] == 0 && 40320 720 24 2 a[1] a[3] a[5] a[7] ----- - ---- + ---- - ---- + a[9] == 0 && 40320 720 24 2 -a[0] a[2] a[4] a[6] a[8] ------- + ----- - ---- + ---- - ---- + a[10] == 0 3628800 40320 720 24 2
Input := Solve[%]
Output = 50521 277 {{a[9] -> 0, a[10] -> -------, a[7] -> 0, a[8] -> ----, 3628800 8064 61 5 a[5] -> 0, a[6] -> ---, a[3] -> 0, a[4] -> --, a[1] -> 0, 720 24 1 a[2] -> -, a[0] -> 1}} 2
Input := Sum[a[k] x^k, {k,0,10}] /. %
Output = 2 4 6 8 10 x 5 x 61 x 277 x 50521 x {1 + -- + ---- + ----- + ------ + ---------} 2 24 720 8064 3628800
Input := Series[1/Cos[x],{x,0,10}]
Output = 2 4 6 8 10 x 5 x 61 x 277 x 50521 x 11 1 + -- + ---- + ----- + ------ + --------- + O[x] 2 24 720 8064 3628800
Input := 1/Series[Cos[x],{x,0,10}]
Output = 2 4 6 8 10 x 5 x 61 x 277 x 50521 x 11 1 + -- + ---- + ----- + ------ + --------- + O[x] 2 24 720 8064 3628800
Input := Series[x+Sin[x],{x,0,10}]
Output = 3 5 7 9 x x x x 11 2 x - -- + --- - ---- + ------ + O[x] 6 120 5040 362880
Input := Plot[x+Sin[x],{x,-4Pi,4Pi}]
Output = -Graphics-
Input := InverseSeries[%49]
Output = 3 5 7 9 x x x 43 x 223 x 11 - + -- + ---- + ------- + -------- + O[x] 2 96 1920 1290240 92897280
Input := Normal[%57]
Output = 3 5 7 9 x x x 43 x 223 x - + -- + ---- + ------- + -------- 2 96 1920 1290240 92897280
Input := Plot[%59,{x,-10,10},PlotRange->{-20,20}]
Output = -Graphics-
Input := ParametricPlot[{x+Sin[x],x},{x,-10,10},PlotRange->{-20,20}]
Output = -Graphics-
Input := Show[%63,%64]
Output = -Graphics-