I am trying to come up with a list of series and Taylor series I should probably know before I take my qualifying exam in august. Here is what I got, please let me know if one of them is wrong or if you have something you think I should add
$(1)$ Geometric series: $\sum_{k = 0}^\infty ar^k = \frac{a}{1 - r}$ if $|r| < 1$.
$(2)$ Alternating Series: $\sum_{n = 1}^\infty (-1)^{n + 1}a_n$ and this converges iff $a_n$ converges to $0$.
$(3)$ Binomial Theorem: $(a + b)^n = \sum_{r = 0}^{n} \binom{n}{r} a^rb^{n - r}$.
$(4) e^t = \sum_{k = 0}^\infty \frac{t^k}{k!}$
$(5) 2^n = \sum_{r = 0}^n \binom nr$
$(6) \frac{1}{(1 - t)^r} = \sum_{k = r}^\infty \binom{k-1}{r-1} t^{k-r}$ For $|t|< 1$
$(7) \frac{t^m}{1 - t} = \sum_{i = m}^\infty t^i$ For $|t|< 1$
$(8) \frac{1 - t^{m +1}}{1 - t} = \sum_{i = 0}^m t^i$ for $|t| < 1$
$(9) \sin x= \sum_{n=0}^{\infty}\frac{(-1)^n\cdot x^{2n+1}}{(2n+1)!}$
$(10) \cos x=\sum_{n=0}^{\infty} \frac{(-1)^n\cdot x^{2n}}{(2n)!}$
taylor series for sinh, cosh, tan$^{-1}$