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Compute the integral of dz/z , where γ : [0, 2π] → C is a circle γ(t) = re$^{it}$ , as follows. Let P˙ be a partition where γ(tk)’s divide circle into equal arcs, tk = 2kπ/n (k = 0, 1 . . . , n); γ(τk) are the midpoints of corresponding arcs, τk = (2k − 1)π/n (k = 1, . . . , n). Find S(f,P˙) and its limit as |P| →˙ 0.

I know that the first step is to write the sum out as $\lim_{P\to 0}$ $\sum_1^n f(τk)(t$$_k$$ -t$$_{k-1}$$)$ which is $\lim_{P\to 0}$$\sum_1^n \frac {n}{2k-1}* r(e$$^{i2kpi/n}$$ - e$$^{2(k-1)ipi/n})$

I have no idea what to do from here. I know by integrating the normal way that the answer is 2*pi*i

*EDITED: the previous version said z/dz, but that was a typo. Was supposed to be dz/z.

Thanks

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    I don't know what an integral with $dz$ in the denominator means. – 2017-02-27
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    Everybody loves Reimmann... – 2017-02-27
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    Hint: this looks like the partial sum of the Taylor expansion of $\ln(1-x)$. – 2017-02-27
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    @GEdgar my bad. It was supposed to be dz/z – 2017-02-27

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