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Calculus/regarding summation to infinity

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Question
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.

Answer
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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