Geometric Series
$1 + \cfrac{1}{2} + \left( \cfrac{1}{2} \right)^2 + \left( \cfrac{1}{2} \right)^3 + \ … \ = \sum\limits_{k=0}^\infty \left( \cfrac{1}{2} \right)^k$ It converges to 2
$1 + x + x^2 + x^3 + \ … \ = \sum\limits_{k=0}^\infty x^k = \cfrac{1}{1 - x}$
- let $y = 1 + x + x^2 + x^3 + \ …$ / multiply by $-x$:
- $-xy = -x\, (1 + x^2 + x^3 + \ …) = -x -x^2 - x^3 - \ …$
- $y - xy = (1 + x^2 + x^3 + \ … \ ) -x -x^2 - x^3 - \ … = 1$
- $y - xy = y\, (1 - x) = 1$
- so $y = \cfrac{1}{1 - x}$
| But it holds only for $ | x | < 1$ |
Suppose $x = 1$
- $1 + 1 + 1 + \ … \ \to \infty$
-
$y = \cfrac{1}{1 - x} = \cfrac{1}{0}$ not defined
- $x = -1$
- $1 - 1 + 1 - 1 + 1 - \ … \ = 0$?
- let’s try to group terms
- $(1 - 1) + (1 - 1) + (1 - 1) + \ … \ = 0$
- $1 - (1 - 1) - (1 - 1) - (1 - 1) - \ … \ = 1$
- maybe take the average? $\cfrac{0 + 1}{2} = 0.5$
- but none of these is correct: this series does not converge