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,

= lim_{n->infinity} 1 + 1/2 + 1/3 + .... + 1/n - ln n

exists.

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

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