1. 기초수학

 

1. logarithms

$ X = a^b, \ \ b = log_aX \\$ $ log \ ab = log \ a + log \ b \\$ $ log \ \frac a b = log \ a - log \ b \\$ $ log \ a^b = b \ log \ a \\$ $ log_an = \frac {log_bn} {log_ba} = \frac {log \ n} {log \ a} \\ $ $ log_aa = 1, for \ all \ a > 0 \\ $ $ log_a1 = 0, for \ all \ a > 0 \\$

2.summation

  2-1. Arithmatic sum

$ \sum_{i=1}^n i = 1 + 2 + 3 ... + i \\ $ $ \sum_{i=1}^n i = \frac {n(n+1)} 2 \\ $ $ \sum_{i=1}^n i^2 = \frac {2n^3 + 3n^2 + n} 6 \\ $

2-2. Geometric sum

$ \sum_{i=0}^n a^i = 1 + a^1 + a^2 + ... a^{n-1} + a^n \\ $ $ S = \sum_{i=0}^n a^i = \frac {a^{n+1} - 1} {a-1} \\ $ $$ S = 1 + a^1 + a^2 + ... + a^{n-1} + a^n \\ aS = a^1 + a^2 + a^3 + ... + a^n + a^{n+1} \\ ----------------- \\ (1-a)*S \ =\ 1 - a^{n+1} \\ S = \ \frac {a^{n+1} - 1} {a-1} \\ lim_{x \to \infty}\ S =\sum_{i=0}^\infty\ a^i = \frac {1} {1-a} \ (when, a < 1 ) $$

 

3. Mathematical Proofs

• 직접 증명
• 간접 증명 or 귀류법
• 귀납법