Taylor and Maclaurin Theorem
December 8, 2016
Categorised in: 1st Year Maths 1
Maclaurin’s Theorem:
$f(x) = f(0) + xf'(0) + x^2/{2!} f”(0) + x^3/{3!} f”'(0) +…+ x^n/{n!} f^n(0) +…$
Standard Expansions:
$sinx = x – x^3/{3!} + x^5/{5!} – x^7/{7!} +…$
$cosx = 1 – x^2/{2!} + x^4/{4!} – x^6/{6!} + …$
$tanx = x + x^3/3 + 2/{15} x^5 + {17}/{315} x^7 + …$
$e^x = 1 + x +x^2/{2!} + x^3/{3!} + x^4/{4!} + …$
$ sinhx = x + x^3/{3!} + x^5/{5!} + x^7/{7!} +… $
$coshx = 1 + x^2/{2!} + x^4/{4!} + x^6/{6!} + …$
$tanhx = x – x^3/3 + 2/{15} x^5 – {17}/{315} x^7 + …$
$log(1+x) = x – x^2/2 + x^3/3 – x^4/4 + …$
$ log(1-x) = -x – x^2/2 – x^3/3 – x^4/4 – … $
$ (1+x)^{-1} = 1-x + x^2 – x^3 + x^4 – … $
$ (1-x)^{-1} = 1 + x + x^2 + x^3 + x^4 + … $
$ tanh^{-1}x = 1/2 log({1+x}/{1-x}) $
$ (1+x)^n = 1 + nx + {n(n-1)}/{2!} x^2 + {n(n-1)(n-2)}/{3!} x^3 +… $
Taylor’s Theorem:
$f(a+h) = f(a) + hf'(a) + h^2/{2!} f”(a) +…+h^n/{n!} f^n(a) + …$
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