1 sequences

1.0.0.1 arithmetic

Difference is denoted with d, such that d = a_{n+1} - a_n

Sum: S_n = \frac{2a_1 + (n-1)d}{2}n

n-th term: a_n = a_1 + (n-1)d

Relation: a_n = \frac{a_{n-1} + a_{n+1}}{2}

1.0.0.2 geometric

Quotient is denoted with q, such that q = \frac{a_{n+1}}{a_n}

Sum: S_n = a_1\frac{1 - q^n}{1 - q}

n-th term: a_n = a_1q^{n-1}

Relation: a_n^2 = a_{n-1}a_{n+1}

1.0.0.3 bound

|a_n| < M

1.0.0.4 limits

A sequence is convergent if it has a limit otherwise it is divergent

L - limit of a sequence

\epsilon -

then: |a_n - L| < \epsilon

• \lim(a_n \pm b_n) = \lim a_n \pm \lim b_n
• \lim(a_nb_n) = \lim a_n \lim b_n
• \lim(\frac{a_n}{b_n}) = \frac{\lim a_n}{\lim b_n}
• \lim(c \cdot a_n) = c\lim a_n

1.0.0.5 subsequences

a_n converges to L if all of its subsequences converge to L

let a_n = (-1)^n then a_{2n} = (-1)^{2n} and a_{2n-1} = (-1)^{2n-1} are subsequences of a_n. They have different limits therefore a_n has no limit.

1.0.0.6 sandwich theorem

a_n \leq b_n \leq c_n

if \lim a_n = \lim c_n = L then \lim b_n = L

1.0.0.7 Euler limit

\lim \limits_{n \to \infty} (1 + \frac{x}{n})^n = e^x

1.0.0.8 L’Hopital rule

Iff \lim_{x \to c} f(x) = \lim_{x \to c} g(x) = 0 or \lim_{x \to c} f(x) = \pm \lim_{x \to c} g(x) = \pm \infty then \lim_{x \to c}\frac{f(x)}{g(x)} = \lim_{x \to c}\frac{f'(x)}{g'(x)}

implicit usage:

• 0 \cdot \infty: f \cdot g = \frac{f}{\frac{1}{g}}
• \infty - \infty: f - g = \frac{\frac{1}{g} - \frac{1}{f}}{\frac{1}{fg}}
• 0 \cdot \infty: f \cdot g = \frac{f}{\frac{1}{g}}
• 0^0, \infty^0, 1^\infty: f^g = e^{g \ln f}