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Cake day: June 6th, 2023

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  • The lyrics of the rest of the song aren’t that great

     Strapped in the chair of the city's gas chamber
         Why I'm here, I can't quite remember
    The Surgeon General says it's hazardous to breathe
               I'd have another cigarette
    But I can't see, tell me who you're gonna believe
    
            Take me down to the Paradise City
    Where the grass is green and the girls are pretty
    

    So either the chorus is pure sarcasm, or his day to day to life is so unbearable that he genuinely wishes to go back to a time/place where the grass is green, and a time when he was still attracted to women











  • For (1), we started with the Maclaurin series 1/x to get us familiar with the idea of differential expansions, and then we moved to Taylor to derive expansions of some common functions like cos and sin:

    cos(x) = 1 - x2/2! + x4/4! - …
    sin(x) = x - x3/3! + x5/5! - …

    We now start with the definition of ex Taylor expansion, and proceed to do some substitutions:

    ex = 1 + x + x2/2! + x3/3! + … + xn/n!

    We can then substitute in: x=iθ (remembering that i2 = -1) to get

    e = 1 + iθ - θ2/2! - iθ3/3! + θ4/4! + iθ5/5! + … etc…

    If we group by real and complex, we can arrange the above as:

    e = (1 - θ2/2! + θ4/4! + … ) + i(θ - θ3/3! + θ5/5! + … )

    You should now realise that the left part resembles the expansion of cos(θ), and the right part resembles sin(θ). That is:

    e = cos(θ) + i sin(θ)

    Finally, we substitute in θ = π

    e = cos(π) + i sin(π)

    And we know that cos(π) = -1, and that sin(π) = 0, meaning that we end up with

    e = -1 + i 0

    or

    e + 1 = 0

    The teacher got excited because it is literally one of the most beautiful mathematical statements you can get, that connects five universal identities under a single statement: 0, 1, e, i, and π – and does so using 3 different operators (times, power, plus).

    For (2), I’m still waiting as I think it’s currently holding the world together by sheer mass alone