quartz/content/Obsidian Vault/Series.md

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#math/calculus

definition of series

#card/reverse The sum of a sequence. Use the \Sigma_{n=1}^\infty notation. ^1652969721739

partial sum

It's a series that converges to a value. So, you're doing part of the sum at a time.

algebraic vs geometric

Algebraic series ::: the function uses n in the function as a normal coefficient. ^1652972454834

Geometric series

#card/reverse the function in the series uses n as an exponent. Thus, the growth is big very cool. ^1652972485160

a is the constant, r is rate

Formula for value at any n: ==$S_n=\frac{a(1-r^{n})}{1-r}$== ^1652972574160

Formula for value at infinity(a is the start): ==$S_n=\frac{a}{1-r}$== ^1653167523226

nice little formula

\Sigma_{n=1}x^{n} and $|x|<1$ $a=x$ $= \frac{x}{1-x}$ if: $n=0$ $a=1$ = \frac{1}{1-x}

sum properties

• \Sigma c\cdot a_{n} = c \cdot \Sigma a_{n}
• \Sigma (a_{n}+b_{n}) = \Sigma a_{n}+ \Sigma b_n
• \Sigma (a_{n}-b_{n}) = \Sigma a_{n} - \Sigma b_n