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QUESTION: >>>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.

I could not understand the above point, wherever we stop in the infinite series of 1+1-1+1-1 ... to infinity will either be 1 or 2, why the mathematicians say that it is 0.5.

Also another bizarre claim which mathematicians say 1+2+3+....to infinity is -1/12. How the summation of positive numbers leads to negative number. It simply makes no sense to me. I think we should not perform arithmetic operations (which they do while deriving these bizarre results) should not be done for infinite series.

Please explain.

ANSWER: The series 1+1-1+1-1 is not related to the one you're suggesting at the start.

The theorem to note is that if |d|<1, then the summation from n=0 to n=infinity of d^n

is 1/(1-d). The series given by 1-1+1-1 is a power series gotten d=-1 and that is not in the range (-1,1), for that range does not include the endpoints.

The bizarre claim about the summation of all positive numbers being 1/12 or the summation of positive numbers being negative - both are not true. It sounds like those are even taken from a joke on numbers or are the incomplete set.

I use to know a rather elegant proof that came down to 0 = 1. It was obviously in error somewhere, but the error wasn't seen at first. Upon close inspection, it could be seen that the equation was divided by 0. Here it is:

Take n=1.

Clearly, when both sides are multiplied by n, this gives n^2 = n.

When we subtract one from both sides, we get n^2 - 1 = n - 1,

If we factor the left side, we get (n+1)(n-1) = n - 1.

Dividing both sides by n-1 gives n+1 = 1.

Subtract 1 from both sides and get n = 0.

Now we started out assuming that n = 1, but proved n = 0.\

That is from division by 0, for n-1 = 0.

In a similar fashion, the 2nd problem you gave me must be incomplete,

for what was said is false.

---------- FOLLOW-UP ----------

QUESTION: >>It sounds like those are even taken from a joke on numbers or are the incomplete set.

No. It is one of the accepted summation by mathematicians. Please see below:

http://en.wikipedia.org/wiki/Ramanujan_summation

http://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF

The address en.wikipedia.org/wiki/Ramanujan_summation was not found.

The address en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF

stated that if the summation was assumed to have an answer, that answer could be shown to be -1/12, but clearly that summation does not have an answer. This is because an element n can't be found for an epsilon-delta proof.

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