Madhava–Leibniz Series
Poetry of Logical Ideas
"It has long been an axiom of mine that the little things are infinitely the most important." ~Sherlock Holmes (A Case of Identity)
Madhava–Leibniz Formula for
In mathematics, the Leibniz formula for
The Fascinating
But how do we get here?
Let’s take a look at the series,
instead of this
Now, our job comes down to evaluating a complex logarithm. We know any complex number can be represented in its polar form i.e.,
So, we need to figure out what’s
In this case, we have
Therefore,
So, we have just replaced the value of
If you string this all the way back to the beginning in
I think this is a nice way to elaborate but it would be better if you do it yourself in pen and paper.
Prove that
One can solve this problem by resolving
Hint
Encounter the problem as it is shown in Resolve
Into Factors. Till equation there will be no change.As you are willing to resolve
instead of . Replace by in equation . and you will see something like the equation below.In equation
the last factor and the last but one-factor and so on.Hence by combining the first and the last factors, the second and the last but one factors and so on, the equation
becomes,Finally, use the limit laws as it’s shown in resolution into factors of
Now, we will solve the problem which is mentioned here. If you resolve
Let’s play with some integral !
Conclusion
There are lots of fascinating fact related to this
See Also
References
Roy, Ranjan (1990). “The Discovery of the Series Formula for π by Leibniz, Gregory and Nilakantha” (PDF). Mathematics Magazine.63(5): 291–306. doi:10.1080/0025570X.1990.11977541.
Ian G. Pearce (2002). Madhava of Sangamagramma. MacTutor History of Mathematics archive. University of St Andrews.
Horvath, Miklos (1983). “On the Leibnizian quadrature of the circle” (PDF). Annales Universitatis Scientiarum Budapestiensis (Sectio Computatorica). 4: 75–83.
Debnath, Lokenath (2010), The Legacy of Leonhard Euler: A Tricentennial Tribute, World Scientific, p. 214, ISBN 9781848165267.
Plofker, Kim (November 2012), “Tantrasaṅgraha of Nīlakaṇṭha Somayājī by K. Ramasubramanian and M. S. Sriram”, The Mathematical Intelligencer, 35 (1): 86–88, doi:10.1007/s00283-012-9344-6, S2CID 124507583.