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Hi,

I read that the summation of series : 1+1-1+1-1 ... to infinity is 1/2. The proof arrived is :

Assume,

S1= 1+1-1+1-1 ... to infinity

S = 1-1+1-1+1... to infity

S1= 1 + S

Therefore, S1-S= 0 + 2-2+2-2+2... to infity

S1-S = 2(1-1+1-1+... to inifty)

1= 2S1

There S1=1/2.

I find that the above way of finding the summation to be incorrect. The addition of 1+1-1+1.. to infinity will either be 2 or 1 but never less than 1. Why mathematicians make such bizarre claims by playing with infinite series like the above. Please explain.

Also 1/3+ 1/3+ 1/3 =1

1/3=0.3333

Therefore 0.3333+0.3333+0.3333=1

Therefore can we say 0.9999 = 1, if not, please explain where the math went wrong?

regards.

For S1 and S, it can be shown that S1-S= 0 +2-2 +2-2 +2-2 indefinitely.

Using that, it can be seen that S1-S=0.

However, it can also be shown that neither S nor S1 really has a limit.

For a series to have a limit, then given any value d, there must be some n such that for any number greater than n, all of the values must be within d of what the answer is.

If we take d = 0.4 (or any number less than 1), clearly no matter how large a value of n we choose, the rest of the terms won't be closer to the proposed limit than that.

For 1/3 = 0.3333, it is really 1/3 = 0.3333... with an infinite number of 3's.

The sum of 1/3 + 1/3 + 1/3 = 0.99999... with and infinite number of 9's.

The value of that is 1, for no matter how small a difference d I choose,

I can find a number n where all of the terms past n are within d of 1.

If d=0.1, then clearly |.33 + .33 + .33 - 1| = .01 < 0.1.

If d=0.0005, then clearly |.3333 + .3333 + .3333 - 1| = .00001 < 0.0005.

In this way, we can say that 0.99999... = 1, for the 9's never stop.

That is just like saying 1/2 + 1/4 + 1/8 + 1/16 + ... + 1/2^n + ... = 1.

Doing it in Excel, that is what we get, as follows:

1/2^n, sum

0.5 0.5

0.25 0.75

0.125 0.875

0.0625 0.9375

0.03125 0.96875

0.015625 0.984375

0.0078125 0.9921875

0.00390625 0.99609375

0.001953125 0.998046875

0.000976563 0.999023438

0.000488281 0.999511719

0.000244141 0.999755859

0.00012207 0.99987793

6.10352E-05 0.999938965

3.05176E-05 0.999969482

1.52588E-05 0.999984741

7.62939E-06 0.999992371

3.8147E-06 0.999996185

1.90735E-06 0.999998093

9.53674E-07 0.999999046

4.76837E-07 0.999999523

2.38419E-07 0.999999762

1.19209E-07 0.999999881

5.96046E-08 0.99999994

2.98023E-08 0.99999997

1.49012E-08 0.999999985

7.45058E-09 0.999999993

3.72529E-09 0.999999996

1.86265E-09 0.999999998

9.31323E-10 0.999999999

4.65661E-10 1

There is still a difference out to 11 places, but however many digits we need,

we can get, so it really is 1.

Calculus

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