The problem which put Euler on the path of immortality was the BASEL PROBLEM solved in 1735, 91 years since it was proposed . Succinctly put Euler's proof was ;

Using this result to obtain a recursion for pi in Python !

At 500 terms the result is 3.13968..... not too far from the sinister 3.1415.....The recursions are good till about 995 terms (3.14063326091) after which one comes across .... RuntimeError: maximum recursion depth exceeded

Similar programs may be devised on more series as given by Euler, as .....

RECURSIONS FOR e

e the base of natural logarithm is yet another important constant in mathematics .... and once again related to Euler .....

A python program on this recursion is designed via the factorial function ....

A python program on this recursion is designed via the factorial function ....

The limit of recursion is around n = 170 which yields a very accurate value of e.

Thus in a nutshell .....

Thus in a nutshell .....

"Read Euler, read Euler, he is the master of us all."

-- Pierre-Simon Laplace

REFERENCES

(1) Recursive functions for phi and root two

(2) Other pi recursions in Python

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