Разложение в ряд Тейлора

Формулы разложения по Тейлору:

e^x = 1 + x + \frac{{x^2 }}{{2!}} + ...\frac{{x^n }}{{n!}} + o(x^n )\\ \frac{1}{{\sqrt {1 - x} }} = 1 + \sum\limits_{k = 1}^n {\frac{{(2k - 1)!!}}{{2^k k}}} x^k + o(x^n ) \\ \frac{1}{{1 + x^2 }} = \sum\limits_{k = 0}^n {( - 1)^k x^{2k} + o(x^{2n + 1} )} \\ (1 + x)^a = 1 + ax + \frac{{a(a - 1)}}{{2!}}x^2 + ... + \frac{{a(a - 1)...(a - n + 1)}}{{n!}}x^n + o(x^n )\quad \\ \ln (1 - x) = - x - \frac{{x^2 }}{2} - \frac{{x^3 }}{3} - ... - \frac{{x^n }}{n} + o(x^n ) \\ \ln (1 + x) = x - \frac{{x^2 }}{2} + \frac{{x^3 }}{3} + ... + \frac{{( - 1)^{n - 1} x^n }}{n} + o(x^n ) \\ \ln (a - bx) = \ln a - \sum\limits_{k = 1}^n {\frac{1}{k}\left( {\frac{b}{a}} \right)^k x^k } + o(x^n ) \\ \sin x = x - \frac{{x^3 }}{{3!}} + \frac{{x^5 }}{{5!}} + ... + \frac{{( - 1)^n x^{2n + 1} }}{{(2n + 1)!}} + o(x^{2n + 2} ) \\ \cos x = 1 - \frac{{x^2 }}{{2!}} + \frac{{x^4 }}{{4!}} + ... + \frac{{( - 1)^n x^{2n} }}{{(2n)!}} + o(x^{2n + 1} ) \\ tgx = x + \frac{{x^3 }}{3} + \frac{{2x^5 }}{{15}} + o(x^6 ) \\ ctgx = \frac{1}{x} - \frac{x}{3} - \frac{{x^3 }}{{45}} + o(x^3 ) \\ \arcsin x = x + \sum\limits_{k = 1}^n {\frac{{(2k - 1)!!}}{{2^k k!}}x^{2k + 1} + o(x^{2n + 2} )} \\ arctgx = \sum\limits_{k = 0}^n {( - 1)^k \frac{{x^{2k + 1} }}{{2k + 1}} + o(x^{2n + 2} )} \\ shx = x + \frac{{x^3 }}{{3!}} + \frac{{x^5 }}{{5!}} + ... + \frac{{x^{2n + 2} }}{{(2n + 2)!}} + o(x^{2n + 2} ) \\ chx = 1 + \frac{{x^2 }}{{2!}} + \frac{{x^4 }}{{4!}} + ... + \frac{{x^{2n} }}{{(2n)!}} + o(x^{2n} )