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Let us say there is a function $f(x)$. Let us say that that it has a Fourier (co)sine series representation $$g(x) = \sum_{n=1}^{\infty} a_n\sin(kx) = f(x)$$

I am having difficulty understanding a question that asks me to find the value to which $g(x)$ converges to over an interval $[a, b]$? $g(x)$ is a function, so it cannot converge to a single value over an interval?

bzm3r
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  • We usually write $$ f(x) \sim \sum_{n = 0}^{\infty} [a_{n} \sin(n x) + b_{n} \cos(n x)] $$ to indicate that the formal Fourier series of $ f $ is equal to $$ \sum_{n = 0}^{\infty} [a_{n} \sin(n x) + b_{n} \cos(n x)]. $$ This formal trigonometric series may or may not converge at a given point in the interval on which $ f $ is defined, and even if it converges at a point $ x $, it may not converge to $ f(x) $. – Berrick Caleb Fillmore Mar 01 '15 at 04:28

1 Answers1

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Recall that $\sum_{n=1}^\infty a_n \sin(kx) = \lim_{N\to \infty} \sum_{n=1}^Na_n \sin(kx)$ is indeed a limit, and we say the sum converges if the limit exists. To me, it looks like the question is asking the following:

Suppose $f$'s fourier sine coefficients are $(a_n)_n$. For what values of $x$ does the series converge $\sum_{n=1}^\infty a_n \sin(kx)$ converge? Call this set $D$. Then for what $x \in D$ do we have $f(x) = \sum_{n=1}^\infty a_n \sin(kx)$?

The subtlety here is that in general, the fourier series of a function does not necessarily converge to that function. The question is asking you to figure out when the series actually converges to what you would hope it converges to.

nullUser
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