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How to solve Ramanujan's problem by numerical method and We need to find I0
I0 =
1
1
2
1
3
1
4
1 ...



Define that; I1 =
1
1
3
1
4
1
5
1 ...



And; I2 =
1
1
4
1
5
1
6
1 ...



In the general case; In =
1
1 n 2( ) I
n 1
 Or; In+1 =
1 I
n

n 2( ) I
n

Notice I0 that 0 < I0 < 1
So, We need to find the condition of Ik+1 in the Programming1 when k is the large number
Initial Condition for Programming k 9999
Programming 1; Assume I0 = 0.5
FindValue1 k( ) I
0
0.5
I
n 1
1 I
n

n 2( ) I
n


n 0 kfor
I

I FindValue1 k( )
0
0
1
2
3
4
5
0.5
0.5
0.333
0.5
0.2
...

So that; I
k 1
0.01
We can approximate that; I
k 1
0
Programming 2;
FindValue2 k( ) I
k 1
0
I
n
1
1 n 2( ) I
n 1


n k 0for
I

I FindValue2 k( )
0
0
1
2
3
4
5
0.525
0.452
0.404
0.369
0.342
...

So that, the finally; I
0
0.52513527616098121

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How to solve Ramanujan's problem by numerical method 2

  • 1. How to solve Ramanujan's problem by numerical method and We need to find I0 I0 = 1 1 2 1 3 1 4 1 ...    Define that; I1 = 1 1 3 1 4 1 5 1 ...    And; I2 = 1 1 4 1 5 1 6 1 ...    In the general case; In = 1 1 n 2( ) I n 1  Or; In+1 = 1 I n  n 2( ) I n  Notice I0 that 0 < I0 < 1 So, We need to find the condition of Ik+1 in the Programming1 when k is the large number Initial Condition for Programming k 9999 Programming 1; Assume I0 = 0.5 FindValue1 k( ) I 0 0.5 I n 1 1 I n  n 2( ) I n   n 0 kfor I  I FindValue1 k( ) 0 0 1 2 3 4 5 0.5 0.5 0.333 0.5 0.2 ...  So that; I k 1 0.01 We can approximate that; I k 1 0 Programming 2; FindValue2 k( ) I k 1 0 I n 1 1 n 2( ) I n 1   n k 0for I  I FindValue2 k( ) 0 0 1 2 3 4 5 0.525 0.452 0.404 0.369 0.342 ...  So that, the finally; I 0 0.52513527616098121