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Some Common Series Used

Arithmetic Progression:: ∑_i=kmi=(k+m)∗(m−k)/2=(m2−k2)/2\sum\_{i=k}^m i = (k + m)*(m - k)/2 = (m^2 - k^2)/2; More specific, for all i,∥a_i−1−a_i∥=d,d∈Ri, \|a\_{i-1} - a\_{i}\| = d, d\in \R we have (a_0+a_n)∗(n+1)/2(a\_0 + a\_n)*(n+1)/2

Geometric Series: for r≠1,<1r\ne 1, < 1

  • Finite: ∑_i=0nari=a(1−rn1−r)\sum\_{i = 0}^n ar^i = a(\frac{1 - r^n}{1 - r})
  • Infinite: ∑_i=0∞ari=a1−r\sum\_{i = 0}^{\infty} ar^i = \frac{a}{1 - r}

Binomial Series: (a+b)n=∑k=0nnCk⋅akbn−k(a+b)^n = \sum\limits_{k=0}^n nCk\cdot a^kb^{n-k}

  • infinite: (1+x)n=∑k=0∞(nk)xk(1+x)^n = \sum_{k = 0}^{\infty} {n\choose k}x^k

Multinomial Series: (a1+a2+⋯+al)n=∑k1,k2,…,kln!k1!k2!…kl!∏i=1laiki(a_1 + a_2 + \cdots + a_l)^n = \sum_{k_1, k_2 , \ldots , k_l} \frac{n!}{k_1!k_2!\ldots k_l!} \prod_{i=1}^l a_i^{k_i}

Negative Binomial Series: (a−b)−n=∑k=0∞(−n)Ck⋅akb−n−k(a-b)^{-n} = \sum\limits_{k=0}^{\infty} (-n)Ck\cdot a^kb^{-n-k}

Commmon limitation​

lim⁡n→0an−1n=ln⁡a\lim\limits_{n\to 0} \frac{a^n - 1}{n} = \ln a

lim⁡n→∞(1+axn)n=eax\lim\limits_{n\to \infty} (1+ \frac{ax}{n})^n = e^{ax}