Seemingly contradictory results The following infinite sums produce remarkable results. $1+2+3+4+...=-\frac{1}{12}$ $1-2+3-4 +...=\frac{1}{4}$ So how are these results compatible with the statement; that integers are closed under addition? Analysis and algebra seem to disagree here. > Is this some sort of abuse of notation? Is this an example of _inconsistent_ statements under the logicians meaning of the word _inconsistent_? And are there any other examples of seemingly contradictory statements from history or presently? **Note.** I'm not asking for a proof of the results.

Addition, the operation that integers are closed under, is a _binary_ operation: taking two inputs, so $ a+b $ only. By an induction argument, closure under addition implies closure under finite sums since finite sums are defined in terms of repeated binary sums.

However, none of the ways to define infinite sums are just "repeated application of binary addition" so your examples just show something like "for some of the nonstandard methods for defining the 'sum' of an infinite sequence, the integers aren't closed under that operation".

The fact that $+$ signs are used is potentially confusing here since binary addition is not being used. That's the abuse of notation.