- #1

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Does that equal 0 or 1?

(1-1) + (1-1) + (1-1) + ... = 0 + 0 + 0 = 0

or

1 + (-1 + 1) + (-1+1) + (-1+1) + ... = 1 + 0 + 0 + 0 = 1

- Thread starter tahayassen
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- #1

- 269

- 1

Does that equal 0 or 1?

(1-1) + (1-1) + (1-1) + ... = 0 + 0 + 0 = 0

or

1 + (-1 + 1) + (-1+1) + (-1+1) + ... = 1 + 0 + 0 + 0 = 1

- #2

disregardthat

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Note that it is important what you define as terms (the [itex]a_k[/itex]'s are the terms). 1-1+1-1+1-... is not the same as (1-1) + (1-1) + ...

The first sum has terms 1,-1,1,-1,... and so on, but the second sum has 1-1,1-1,1-1,... that is, 0,0,0,... as terms.

Obviously, if 1-1 = 0 are the terms, the series will converge to 0. The sum is simply 0 + 0 + 0 + ... which converges to 0 in the mathematical sense described above. Your example was 1 + (1-1) + (1-1) + ... which of course converges to 1. But this is

On the other hand, if the terms are 1, -1, 1, -1, ... the series does

Be careful around infinite series. Customary properties such as associativity and commutativity of terms doesn't apply in the same way.

- #3

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Be careful around infinite series. Customary properties such as associativity and commutativity of terms doesn't apply in the same way.

So is his method for finding the sum of that infinite series incorrect?

Thanks for clearing that up by the way.

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- #4

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However, if you insist on assigning a value to this series, the best such value is 1/2. Beware: The techniques used to do this will also say that 1+1+1+1+... = -1/2 and that 1+2+4+8+...=-1.

- #5

disregardthat

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Technically it is incorrect using the definition of convergence I described above. The series 1+2+4+16+... does not converge (it diverges), and summing two infinite series require convergence of both.

So is his method for finding the sum of that infinite series incorrect?

Thanks for clearing that up by the way.

However there are other kinds of summations, see

http://en.wikipedia.org/wiki/1_+_2_+_4_+_8_+_…

which is something we customarily don't use when summing series. But in that context it can in fact be so that 1 + 2 + 4 + 16 + ... = -1. But that's not to say that the series "approach" or "tend to" -1. The methods used in the video are strictly incorrect though.

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