28 Oct '14 20:256 edits

I have a problem with the infinite series:

S = 1 + 2 + 3 + 4 …

equalling -1/12

In one of the links I recently looked at, it said that the sum of the infinite series:

1 + ½ + ½ + ½ + ½ + ½ + ½ …

Is infinity. So surely the sum of the infinite series:

S1 = 1 + 1 + 1 + 1 + 1 + 1 + 1 …

is also infinity?

OK, assuming this is the case, the infinite series:

S = 1 + 2 + 3 + 4 …

can be rewritten as:

1 + (1+1) + (1+2) + (1+3) + (1+4) …

which means it is equal to the sum of the two infinite series:

S1 = 1 + 1 + 1 + 1 + 1 + 1 + 1 …

AND

S = 1 + 2 + 3 + 4 …

i.e. S = S1 + S

But note that S1 = infinity (I assume ) and the previous proof apparently proved S = -1/12. So surely, -1/12 + infinity equals infinity thus S is infinity?

(and this is not to mention the apparent logical contradiction of the values of the two sides of the equation being not equal )

I can only assume I am wrong but, what exactly is the flaw with my above logic?

S = 1 + 2 + 3 + 4 …

equalling -1/12

In one of the links I recently looked at, it said that the sum of the infinite series:

1 + ½ + ½ + ½ + ½ + ½ + ½ …

Is infinity. So surely the sum of the infinite series:

S1 = 1 + 1 + 1 + 1 + 1 + 1 + 1 …

is also infinity?

OK, assuming this is the case, the infinite series:

S = 1 + 2 + 3 + 4 …

can be rewritten as:

1 + (1+1) + (1+2) + (1+3) + (1+4) …

which means it is equal to the sum of the two infinite series:

S1 = 1 + 1 + 1 + 1 + 1 + 1 + 1 …

AND

S = 1 + 2 + 3 + 4 …

i.e. S = S1 + S

But note that S1 = infinity (I assume ) and the previous proof apparently proved S = -1/12. So surely, -1/12 + infinity equals infinity thus S is infinity?

(and this is not to mention the apparent logical contradiction of the values of the two sides of the equation being not equal )

I can only assume I am wrong but, what exactly is the flaw with my above logic?