The idea of this post is to highlight a remarkable result: The sum of all the natural numbers to infinity is:

.

First, we may consider a different sum, , (to ):

,

On first inspection, one might think the value of is (i.e and infinite sum of ‘s) or (i.e plus an infinite sum of ‘s). However, it’s actual value can be determined algebraically. If we write out ,

we find that

.

This rearranges to,

,

which is sensible, it can be thought of as the average of ending the sum at a plus and at a minus .

Secondly, we can consider another sum,

,

that, when added to itself (with the help of visual alignment) reduces to

.

Which is , and gives rise to a new algebraic expression,

,

that may be solved giving a value of ,

.

This result can be used to help us evaluate , if we compute ,

and extract a factor of four from each term,

then rearranging what is left gives,

.

Thus, the sum of all the positive numbers unto and including infinity is not infinity, but .

Note that credit should be given to Numberfile for the videos (a) and (b), where i picked up on this.

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