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mfb

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I guess you are referring to things like this? Forget it. Those calculations are mathematically nonsense, and you can get any result you like if you use those wrong operations. They are not the way those pseudo-limits are defined.I've tried to add, subtract etc, but I couldn't prove it, anyone here could help?

As an example, consider

1+1+1+1+.... = X

Let's add 1 to both sides:

But the left sides of both equations are identical, therefore, X=1+X.

Subtract X:

0=1.

- #3

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I meant "adding" sums, likeI guess you are referring to things like this? Forget it. Those calculations are mathematically nonsense, and you can get any result you like if you use those wrong operations. They are not the way those pseudo-limits are defined.

S= 1-2+3-4+5... And adding S+S, it'll eventually give us 2*S=1-1+1-1+...

That's how we can assign values to these series => S= 1/4 (1-1+1-1+1...= 1/2 )

- #4

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What about ##1 + 4 + 9 + 16 \dots = 1 +(-1 + 1) + 4 + (-5 + 5) + 9 + (-14 + 14) + 16 + (-30 + 30) \dots = 0 + 5 - 5 + 14 - 14 + 30 - 30 + \dots = 0 + 0 + 0 + 0 \dots = 0##

Although:

##1 + 2 + 3 + 4 + 5 \dots = 1 + (-1 + 1) + 2 + (-3 + 3) + 4 + (-7 + 7) + 5 + (-12 + 12) \dots) = 0##

Looks like they all sum to 0.

- #5

mfb

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You can, but those values are meaningless. See the example I put in my post, you can easily make contradictions like that.That's how we can assign values to these series => S= 1/4 (1-1+1-1+1...= 1/2 )

It is not what mathematicians do to assign values to those series!

- #6

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I wrote that I add sums, in general, mathematicians do that, but in this example you added 1 to an infinite series that's wrong, infinity + 1 = infinity.As an example, consider

1+1+1+1+.... = X

Let's add 1 to both sides:

1+1+1+1+... =1+X

But the left sides of both equations are identical, therefore, X=1+X.

Subtract X:

0=1.

As written in that wikipedia article mathematician 'assign' the values to series, maybe equal is much said, but how to assign 0 to the 1+4+9+... would be a better formulation of the question.

This method of assigning is used in many papers(including string theory).

A video from Numberphile would explain visually what I mean:

- #7

ShayanJ

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I don't remember in which, but he does explain assigning values to divergent series.

- #8

mfb

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Not in the way you wrote here, no.I wrote that I add sums, in general, mathematicians do that

That is my point, the operations you do with the series are wrong.but in this example you added 1 to an infinite series that's wrong

There are various methods, in this particular case it is Ramanujan summation. More options (which can lead to different answers for the same series) are listed here.As written in that wikipedia article mathematician 'assign' the values to series, maybe equal is much said, but how to assign 0 to the 1+4+9+... would be a better formulation of the question.

- #9

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How would be a correct way to write it?Not in the way you wrote here, no

There are various methods, in this particular case it is Ramanujan summation.

I would like to know the steps he applied to be able to assign 0 to the middle series.

[PLAIN]http://QUOTE="mfb, [Broken] post: 5498284, member: 405866"]Not in the way you wrote here, no[/QUOTE] [/PLAIN]

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- #10

mfb

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See the section above, "Summation".I would like to know the steps he applied to be able to assign 0 to the middle series.

- #11

Mark44

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This misuse of divergent series has been discussed numerous times here at PF.Those calculations are mathematically nonsense

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