I just need to make sure that I've got this analysis right:(adsbygoogle = window.adsbygoogle || []).push({});

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?

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# Infinite Series

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