Mikroharjoitus 6

Tässä yksi tapa eli minun tapani ratkaista kuudennet mikroharjoitukset.

Markko

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Teht. 1

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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

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Teht. 2

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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-

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Teht. 3

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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-

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Teht. 4

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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-

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Teht. 5

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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

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Teht. 6

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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-