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The Taylor Mac­Laur­in Expan­sion enables some of the most widely util­ised power series approx­im­a­tions in real world phys­ics. Their deriv­a­tions are often over­looked. In this series I attempt to rem­edy this issue! Essen­tially this involves assum­ing that we can expand all (well behaved) func­tions as infin­ite power series. Although that seems to be quite an assump­tion, it works for most equa­tions we are likely to deal with. There­after we take the first or second (of the infin­ite num­ber) terms and these are the Taylor Expan­sion Approx­im­a­tion of the func­tion. When we expand the func­tion around the ori­gin we speak of the Mac­Laur­in Series Expansion.

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Video # Video Tutori­al Title Remarks
1 Bino­mi­al Theorem A deriv­a­tion
2 Deriv­a­tion of Taylor and Mac­Laur­in Series Expansions Two of the most use­ful approx­im­a­tions for real world physics
3 Taylor / Mac­Laur­in Series (1+x)n See why (1+x)(1/2) ~= 1+x/2
4 Taylor / Mac­Laur­in Series for Cos(x) Without this expan­sion we can’t derive Euler’s Equa­tion. eix = Cos(x) + iSin(x)
5 Taylor / Mac­Laur­in Series for Exp(x) Required for Sommerfeld’s Expan­sion for example
6 Taylor / Mac­Laur­in Series for ln(x) Util­ised very very reg­u­larly in physics

 

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