Saturday, February 21, 2009

The Euler-Mascheroni constant

We saw before that the natural logarithm function had a McLaurin expansion,

ln (1+x) = x - x^2/2 + x^3/3 - x^4/4 +..... ---------------------(2)

which was valid absolutely within the unit circle. Outside this we will begin seeing a discrepancy between the sum of the series on the right hand side of (2) and ln (1+x). Using Taylor's theorem the error term can actually be calculated and the series treated as an approximation to the function. Then, as x grows large without bound, what becomes of this discrepancy?

We define,
gamma = lim_{n->infinity}  1 + 1/2 + 1/3 + .... + 1/n   - ln n
We immdiately see by the bounds imposed by the Taylor series (2) that gamma 

Since gamma exists we in turn have another way to prove that the harmonic series diverges given the natural logarithm does!

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