I have a series of terms as follows:
$$e^{6x\pi.0} + e^{6x\pi.2} + e^{6x\pi.4} + e^{6x\pi.6}$$
Obviously the first term is just 1 but is there a way to specify the terms in one single term or shorten it somehow other than just 1 + ...?
i is an unknown in the expression
Answer
If the ".k" is multiply by $k$ then:
$$S=1+e^{6x\pi 2}+e^{6x\pi 4}+ e^{6x\pi 6}$$
$$Se^{6x\pi 2}=e^{6x\pi 2}+e^{6x\pi 4}+e^{6x\pi 6}+ e^{6x\pi 8}$$
Substract the two expressions:
$$Se^{6x\pi 2}-S=(e^{6x\pi 2}+e^{6x\pi 4}+e^{6x\pi 6}+ e^{6x\pi 8})-(1+e^{6x\pi 2}+e^{6x\pi 4}+ e^{6x\pi 6})$$
The only terms which remains are $1$ and $e^{6x\pi 8}$:
$$S(e^{6i\pi 2}-1)=e^{6i\pi 8}-1$$
$$S = \frac{e^{6i\pi 8}-1}{e^{6i\pi 2}-1}$$
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