Mixing the two expressions from my earlier two blog posts, one can get
You may note that we are dealing with infinite series here. (-n) and (-n+1) follow one another in one step. And combination (n,n-1) and combination (n+1,n) too. So that is the beauty here. And the result is 1.
You may know that e to the power of i into theta has similar properties. cos theta and sine theta follow one another. But the phase difference is 90 degrees. Step of i to the power n.
Complex numbers are not just a+ib sorts. That is just the beginning. Electrical engineering supports this because magnetism and electricity have a 90 degree phase difference. i.
So complex numbers are much more complex than you would have learnt. Riemann zeta function, Dirichlet beta functions are functions dealing with complex numbers and they have a simple form which is suitable to numbers and number theory.
Did you learn something useful here ?
It is logarithmic series.
ReplyDeleteThere may be nothing great about the expression and title may be misleading you.
My intention is not to mislead you or confuse you.