meow

<3

Pure <3

Here's another:

.999~ = sigma(.9*[.1]^[n-1]).

.999~ = .9 + .09 + .009 + .0009 + .00009 + .000009 .... (ad infinitum)

Let a, 1st term = .9

Let r, common ratio = 10 ^ -1

Infinite geometric progression: a + ar + arÂ²... (ad infinitum)

Sum to infinity = a / (1 - r) = .9 / (1 - .1) = .9 / .9 = 1

.999~ = 1

http://www.geocities.com/admiralmussina/SigmaNinev2.rtfsigma(n = 1, n -> inf.) 9/(10^n) = Definition of geometric series.

9 * sigma(n = 1, n -> inf.)(1/10)^n = Property of a series.

9 * 1/9 = (r/[1 - r], r = .1)

3/9 = 1/3

.999~ = 1

Let Sequence A = (.9, .99, .999... ad infinitum)

Let Sequence B = (1, 1, 1...)

If sequences A and B were equivalent, then they would have the same limit. In sequence A, the infinith degree holds .999~. For sequence A to be equal to sequence B, then sequence B must have the infinith degree be equal to .999~. Since sequence B holds only 1, then for sequence A and B to be equal, .999~ must equal 1.

Let S = .999~

S = .9 + .09 + .009 + .0009 + .00009 + .000009 .... (ad infinitum)

S = 0.9 + (1/10)S

(9/10)S = .9

1 = S

Sequence A = Sequence B

.999~ = 1

For two numbers to be different, there must be a number between them.

.999~ < N < 1

Find N. Because .999~ has infinite nines, there are no numbers between them, thus N cannot exist. .999~ = 1.

.999~ = .9 + .09 + .009 + .0009 + .00009 + .000009 .... (ad infinitum)

sigma[-1:0 -> inf.](.9 * .10^-1)

omega -> sigma = .9 / (1 - .1) = .9 / .9 = 1

.999~ = 1

.(42)~ = 42/99

.(402)~ = 402/999

.000~ = 0/9

.111~ = 1/9

.222~ = 2/9

.333~ = 3/9

.444~ = 4/9

.555~ = 5/9

.666~ = 6/9

.777~ = 7/9

.888~ = 8/9

.999~ = 9/9

9/9 = 1

.999~ = 1

.333~ = sigma(n = 1, n -> inf.) 3/(10^n) = Definition of geometric series.

3 * sigma(n = 1, n -> inf.) (1/10)^n = Property of a series.

3 * 1/9 = (common ratio, r/[1 - r], r = .1)

3/9 = 1/3

1/3 = .333~

1/3 * 2 = 2/3

.333~ * 2 = .666~

2/3 = .666~

.333~ + .666~ = .999~

1/3 + 2/3 = 3/3

3/3 = .999~

3/3 = 1

.999~ = 1

--- Corollary Proof ---

.999~/3 = .333~

1/3 = .333~

.333~ = .333~

.999~/3 = 1/3

.999~ = 1

If Rolken asks why you didn't get the record, tell him it's because you were getting stoned. He'll understand.