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can somebody please show where $\frac{a_0}{2}$ comes from. I don't understand where the "2" comes from. Thank you in advance. Below is a screenshot from a calculus book:

enter image description here

user577215664
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Jek Denys
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  • It is labeled that way in order to make the formula for the first term related to the average value of the function the series represents, that is $$\frac12 a_0 = \frac12 \frac1p \int_{-p}^p f(t) dt = f_{ave}$$ It is related to symmetry this way as odd functions wont have a constant term ($a_0$ really is essentially just the $n=0$ cosine term). – Triatticus Feb 16 '18 at 02:23
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    ok, so far I get that $a_0$ is the cos(0) term, when the first sum is incremented once. But! When I increment the sum by one I don't have 1/2 anywhere :( i.e.: $$\sum_{n=0}^\infty a_n \cdot cos(\frac{n \pi x}{p}) \rightarrow a_0+\sum_{n=1}^\infty a_n \cdot cos(\frac{n \pi x}{p})$$ – Jek Denys Feb 16 '18 at 03:24
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    Nothing stops you from redefining the constant term to absorb the half, some authors do this, the remaining coefficients depend on n in nontrivial ways and the most general way to write them is $a_n$ or $b_n$ – Triatticus Feb 16 '18 at 03:42

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