CVEL1

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Evaluate \sum _{{k=0}}^{{100}}e^{{kz}}\,

\sum _{{k=0}}^{{100}}e^{{kz}}=1+e^{z}+e^{{2z}}+...+e^{{100z}}={\mathrm  {sum}}\,

If z=2n\pi i\, then {\mathrm  {sum}}=1+1+...+1=101\,

If z\neq 2n\pi i\, then multiplying the sum by 1-e^{z}\, reveals:

{\mathrm  {sum}}(1-e^{z})=1-e^{z}+e^{z}-e^{{2z}}+e^{{2z}}+...+e^{{100z}}-e^{{101z}}=1-e^{{101z}}\,

Therefore {\mathrm  {sum}}={\frac  {1-e^{{101z}}}{1-e^{z}}}\,

Also, this can be obtained from the familiar formula

\sum _{{k=0}}^{{100}}x^{{k}}=1+x+x^{{2}}+...+x^{{100}}={\frac  {1-x^{{101}}}{1-x}}\,

where x=e^{z}\,

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