Trigonometry/The summation of finite series

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Problem Statement[edit]

Find a closed form for
.

Note: A 'closed form' is not mathematically defined, but just means a simplified formula which does not involve '...', or a summation sign. In our problem, we should look for a formula that only involves variables , and known operations like the four operations, radicals, exponents, logarithm, and trigonometric functions.

Method 1[edit]

To sum the series

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\displaystyle \sin(a)+\sin(a+b)+\sin(a+2b)+\cdots+\sin(a+(n-1)b)=S} .

Multiply each term by

.

Then we have

and similarly for all terms to

.

Summing, we find that nearly all the terms cancel out and we are left with

.

Hence

.

Similarly, if

then

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\displaystyle C=\cos\left(a+\tfrac{(n-1)b}{2}\right)\cdot\frac{\sin\left(\tfrac{nb}{2}\right)}{\sin\left(\tfrac{b}{2}\right)}} .

Method 2[edit]

Consider the following sum

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\displaystyle s=e^{ai}+e^{(a+b)i}+\cdots+e^{(a+(n-1)b)i}} .

Since is a geometric series with common ratio , we get

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\displaystyle s=\frac{e^{ai}(e^{nbi}-1)}{e^{bi}-1}=\frac{e^{ai}(e^{nbi}-1)}{e^{bi}-1}=\frac{e^{ai}e^\frac{nbi}{2}(e^\frac{nbi}{2}-e^{-\frac{nbi}{2}})}{e^\frac{bi}{2}(e^\frac{bi}{2}-e^{-\frac{bi}{2}})}}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\displaystyle s=\frac{\sin\left(\tfrac{nb}{2}\right)}{\sin\left(\tfrac{b}{2}\right)}\cdot e^{\left(a+\tfrac{(n-1)b}{2}\right)i}}

Therefore,