Analyzing the Argument: Is Algebraic Operations on Infinite Series Valid?

[Gear300]

I just need to make sure that I've got this analysis right:

The argument S = 1 - 1 + 1 - 1 + 1 - ...
then S = (1 - 1) + (1 -1) + (1 -1) + ... = 0 is invalid because it ignores all sum S_n for n not congruent modulo 2 (not even).

The argument S = 1 - 1 + 1 - 1 + 1 - ...
then S = 1 - (1 -1) - (1 - 1) - ... = 1 is invalid because it ignores all sum S_n for n congruent modulo 2 (even).

The argument S = 1 - 1 + 1 - 1 + 1 - ...
then S = 1 - (1 - 1 + 1 - 1 + ...) = 1 - S,
S = 1 - S
S = 1/2 is invalid because S + S (from adding the terms) is the same as S. From this, it seems as though algebraic operations such as addition and multiplication by scalars do not seem to work the same way with infinite series as they do with numbers.

The argument S = 1 + 2 + 4 + 8 ...
Then 2S = 2 + 4 + 8 + 16 ...
2S = S - 1
S = -1 is invalid because (from the third argument) 2S - S is not necessarily S.

What struck me as odd was that even though infinite series, such as these, consist of algebraic operations on Real Numbers (integers in this case), algebraic operations do not seem to work on them in the same way...why is that so?

 

[CRGreathouse]

algebraic operations do not seem to work on them in the same way...why is that so?

Because the series diverges. You can use those rules on series that converge absolutely, but not (without Caesaro, etc. summation) on divergent series.

 

[Gear300]

I see...are there defined operations for diverging series?

 

[g_edgar]

I see...are there defined operations for diverging series?

There are whole BOOKS on the subject ... such as:

G. H. Hardy, Divergent Series, 1929

 

[Gear300]

I see...thanks for the replies.

 

[Elucidus]

Even when the series is conditionally convergent, such as the harmonic series $\sum_{n=1}^{\infty} \frac{1}{n}$, there are still problems with associativity and commutativity of addition and distribution.

There is a theorem (which may be called Riemann's Rearrangement Theorem) that says for any conditionally convergent series there exists a permutation of the terms that will converge to any finite number desired or diverge to infinity or minus infinity.

The only series where the normal algebraic properties hold are those that are absolutely convergent.

--Elucidus

 

[Gib Z]

Its the Riemann Series Theorem: http://en.wikipedia.org/wiki/Riemann_series_theorem