How to solve Ramanujan's problem by numerical method 1
1. How to solve Ramanujan's problem by numerical method and We need to find I0
I0 = 1 2 1 3 1 4 1 ...
Define that; I1 = 1 3 1 4 1 5 1 ...
And; I2 = 1 4 1 5 1 6 1 ...
In the general case; In = 1 n 2( ) 1 n 3( ) 1 n 4( ) 1 ...
In = 1 n 2( ) I
n 1
Or; In+1 =
I
n
2
1
n 2
Notice I0 that 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 = 1.5
FindValue1 k( ) I
0
1.5
I
n 1
I
n
2
1
n 2
n 0 kfor
I
I FindValue1 k( )
0
0
1
2
3
4
1.5
0.625
-0.203
-0.24
...
So that; I
k 1
9.999 10
5
We can approximate that; I
k 1
0
Programming 2;
FindValue2 k( ) I
k 1
0
I
n
1 n 2( ) I
n 1
n k 0for
I
I FindValue2 k( )
0
0
1
2
3
4
5
3
4
5
6
7
...
So that, the finally; I
0
3