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Sum ArcTan Limit (Posted on 2016-05-11) Difficulty: 3 of 5
F(k) denotes the kth Fibonacci Number.

Define:
G(n) = Σk=1 to n ArcTan((F(2k+1)-1)

Find:
Limit G(n)
n → ∞

No Solution Yet Submitted by K Sengupta    
Rating: 4.0000 (1 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
Solution summing the series via computer | Comment 1 of 6
DefDbl A-Z
Dim crlf$


Private Sub Form_Load()
 Form1.Visible = True
 
 
 Text1.Text = ""
 crlf = Chr$(13) + Chr$(10)
 
 pi = Atn(1) * 4
 f1 = 1: f2 = 1: f3 = f1 + f2
 For k = 1 To 100
   term = Atn(1 / f3)
   tot = tot + term
   Text1.Text = Text1.Text & k & "   " & tot & "     " & tot * 180 / pi & crlf
   DoEvents
   f1 = f2: f2 = f3: f3 = f1 + f2
   f1 = f2: f2 = f3: f3 = f1 + f2
 Next k
 
 Text1.Text = Text1.Text & crlf & " done"
  
End Sub


finds

  k        sum               sum * 180/pi
  1   0.463647609000806     26.565051177078
  2   0.661043168850687     37.8749836510982
  3   0.737815060120465     42.2736890060937
  4   0.76721834832447      43.95837332399
  5   0.778453830582433     44.6021190381654
  6   0.782745649724444     44.8480221614361
  7   0.78438499251819      44.9419495846929
  8   0.785011166512743     44.9778267118217
  9   0.78525034373851      44.9915305574138
 10   0.785341701311799     44.9967649607898
 11   0.785376596799798     44.9987643249762
 12   0.785389925690167     44.9995280141399
 13   0.785395016873256     44.9998197174437
 14   0.785396961532153     44.9999311381911
 15   0.785397704325756     44.9999736971295
 16   0.785397988047665     44.9999899531975
 17   0.785398096419791     44.9999961624629
 18   0.78539813781426      44.9999985341913
 19   0.78539815362554      44.9999994401109
 20   0.785398159664911     44.9999997861414
 21   0.785398161971746     44.9999999183133
 22   0.785398162852879     44.9999999687985
 23   0.785398163189441     44.9999999880821
 24   0.785398163317997     44.9999999954478
 25   0.785398163367101     44.9999999982612
 26   0.785398163385857     44.9999999993358
 27   0.785398163393021     44.9999999997463
 28   0.785398163395757     44.9999999999031
 29   0.785398163396802     44.999999999963
 30   0.785398163397202     44.9999999999859
 31   0.785398163397354     44.9999999999946
 32   0.785398163397413     44.999999999998
 33   0.785398163397435     44.9999999999992
 34   0.785398163397443     44.9999999999997
 35   0.785398163397447     44.9999999999999
 36   0.785398163397448     45
 37   0.785398163397448     45
 38   0.785398163397448     45
 39   0.785398163397449     45
 40   0.785398163397449     45
 41   0.785398163397449     45
 42   0.785398163397449     45
 43   0.785398163397449     45
 44   0.785398163397449     45
 45   0.785398163397449     45
 46   0.785398163397449     45
 47   0.785398163397449     45
 48   0.785398163397449     45
 49   0.785398163397449     45
 50   0.785398163397449     45
 51   0.785398163397449     45
 52   0.785398163397449     45
 53   0.785398163397449     45
 54   0.785398163397449     45
 55   0.785398163397449     45
 56   0.785398163397449     45
 57   0.785398163397449     45
 58   0.785398163397449     45
 59   0.785398163397449     45
 60   0.785398163397449     45
 61   0.785398163397449     45
 62   0.785398163397449     45
 63   0.785398163397449     45
 64   0.785398163397449     45
 65   0.785398163397449     45
 66   0.785398163397449     45
 67   0.785398163397449     45
 68   0.785398163397449     45
 69   0.785398163397449     45
 70   0.785398163397449     45
 71   0.785398163397449     45
 72   0.785398163397449     45
 73   0.785398163397449     45
 74   0.785398163397449     45
 75   0.785398163397449     45
 76   0.785398163397449     45
 77   0.785398163397449     45
 78   0.785398163397449     45
 79   0.785398163397449     45
 80   0.785398163397449     45
 81   0.785398163397449     45
 82   0.785398163397449     45
 83   0.785398163397449     45
 84   0.785398163397449     45
 85   0.785398163397449     45
 86   0.785398163397449     45
 87   0.785398163397449     45
 88   0.785398163397449     45
 89   0.785398163397449     45
 90   0.785398163397449     45
 91   0.785398163397449     45
 92   0.785398163397449     45
 93   0.785398163397449     45
 94   0.785398163397449     45
 95   0.785398163397449     45
 96   0.785398163397449     45
 97   0.785398163397449     45
 98   0.785398163397449     45
 99   0.785398163397449     45
100   0.785398163397449     45


So the sum approaches pi/4, as, treated as radians and converted to degrees, it comes out to 45°.

Edited on May 11, 2016, 10:22 am
  Posted by Charlie on 2016-05-11 10:19:23

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