How can one expand the function $f_1(x) = x$ on $(−π, π)$in terms of the functions $\cos nx, n = 0, 1, 2, ...$ and $\sin nx, n = 1, 2, ...,$ I read the below function - $$f_2(x) = a_0 + \sum_{n=1}^{\infty}(a_n \cos nx + b_n \sin nx)$$ is the expansion of the $f_1$, how is that? Main Question: Is $f_1$ equivalent to $f_2$ in some way? If so, plz provide a detail proof which is easily comprehensible. I have attached the excerpt below -
How can one expand the function $f_1(x) = x$ on $(−π, π)$in terms of the functions $\cos nx, n = 0, 1, 2, ...$ and $\sin nx, n = 1, 2, ...,$ in a way so that $$f_2(x) = a_0 + \sum_{n=1}^{\infty}(a_n \cos nx + b_n \sin nx)$$ is the expansion of the $f_1$, how is that? I have asked  related to the same issue but didn't get full answer, I have been told that this is a standard application of Fourier series, $f_1$ and $f_2$ are equivalent in the sense that $f_1(x)=f_2(x)$ for all $x∈(−π,π)$. For any $x$ in this interval, the Fourier series will converge to $f(x)$, but I didn't get any rigorous proof  with elaborate explanation. So, plz provide rigorous proof of $f_1=f_2$ with elaborate explanation as I am new to the subject. Thanks You. I have attached the excerpt below -
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What John did was gave me a cat How to prove that a set is invariant? Why is my profile messed up on a single site and how can I revert it? I was just asking a question on Gardening SE and suddenly my user name there is a25bedc5-3d09-41b8-82fb-ea6c353d75aand my picture is replaced by a Gravatar. My location is also left blank.  Here's a   Curiously, my profiles on all other sites are fine.  I don't have a separate login for SE, I log in using my Google account. Nobody other than me has access to my Google account. I use two step authentication and I can see on "recent devices" in Google that no one but me has accessed my Google account. I only access SE from home, and nobody other than me has used my devices.   I tried to edit my Gardening SE profile back to the way I want it to be and I got the following message:     Oops! 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