Taylor Series of Common Functions
$$
\begin{flalign*}
% 1. e^x
\text{1. } e^x \quad &= 1 + x + \frac{1}{2} x^2 + \frac{1}{6} x^3 + \dotsm & \\
& = \sum_{n = 0}^{\infty} \frac{x^n}{n!} & \\[1ex] % Added a slight vertical space
% 2. sin x
\text{2. } \sin x \quad &= x-\frac{1}{6} x^3 + \frac{1}{120} x^5 + \dotsm & \\
& = \sum_{n = 0}^{\infty} \frac{(-1)^{n} x^{2n + 1}}{(2n + 1)!} & \\[1ex]
% 3. cos x (Note: Added numbering for clarity, using 3 here)
\text{3. } \cos x \quad &= 1-\frac{1}{2} x^2 + \frac{1} {24} x^4 + \dotsm & \\
& = \sum_{n = 0}^{\infty} \frac{(-1)^{n} x^{2n}}{(2n)!} & \\[1ex]
% 4. ln(1+x)
\text{4. } \ln (1+x) \quad &= x-\frac{1}{2}x^2+\frac{1}{3}x^3 + \dotsm & \\
& = \sum_{n = 1}^{\infty} \frac{(-1)^{n + 1}x^n}{n} & \\[1ex] % NOTE: Corrected summation term to x^n/n
% 5. tan x
\text{5. } \tan x \quad &= x+\frac{1}{3}x^3+\frac{2}{15}x^5 + \dotsm & \\
& = \sum_{k = 1}^{\infty} \frac{(-1)^{k – 1} 4^k (4^k-1) B_{2k}}{(2 k)!} x^{2 k-1} & \\[1ex]
% 6. 1/(1-x) (Geometric Series)
\text{6. } \frac{1}{1-x} \quad &= 1 + x + x^2 + x^3 + x^4 + \dotsm & \\
& = \sum_{k = 0}^{\infty} x^k & \\[1ex]
% 7. ln((1+x)/(1-x))
\text{7. } \ln \left(\frac{1 + x}{1-x}\right) \quad &= 2 x + \frac{2}{3} x^3 + \frac{2}{5} x^5+\frac{2}{7}x^7 + \dotsm & \\
& = \sum_{k = 0}^{\infty} \frac{2 x^{2k + 1}}{2k + 1} &
\end{flalign*}
$$
Taylor Series of Less Common Functions
$$
\begin{flalign*}
% 1. sin^{-1} x
\text{1. } \sin^{-1} x \quad &= x + \frac{1}{6} x^3 + \frac{3}{40} x^5 + \dotsm & \\
& = \sum_{k = 0}^{\infty} \frac{(2k)!}{(2^k k!)^2 (2k+1)} x^{2k+1} & \\
& = \sum_{k = 0}^{\infty} \frac{ \Gamma (n + \frac{1}{2}) }{\sqrt{\pi} (2n+1) n!} x^{2k+1} & \\[1ex]
% 2. sec x
\text{2. } \sec x \quad & = 1 + \frac{1}{2} x^2 + \frac{5}{24} x^4 + \frac{61}{720} x^6 & \\
& = \sum_{k = 0}^{\infty} \frac{(-1)^k E_{2n} x^{2n}}{(2k)!} & \\[1ex] % Added a slight vertical space
\end{flalign*}
$$
