Permalink Submitted by Anonymous on January 26, 2016

Consider these two divergent series.

Sp = +1 + 1 + 1 ....
Sn = -1 - 1 - 1 ...

If added together, all terms cancel and Sp + Sn = 0.

However, if one allows the illegitimate preliminary step of "shifting" either series left or right, one can arbitrarily produce a result of any integer value you want to get. The operation of shifting (or of defining a "series" that starts with a finite number of zeros) creates a finite series that becomes the result. The amount of shifting determines the number you get, because the rest of the infinite series will still cancel.

BTW, notice also that Sp + Sn = S1 from the video. So, if one rules out illegitimate cheating steps that artificially create a finite series (e.g. shifting or bracketing), the result is that Sp + Sn = S1 = 0.

## Cheating through shifting a finite number of terms

Consider these two divergent series.

Sp = +1 + 1 + 1 ....

Sn = -1 - 1 - 1 ...

If added together, all terms cancel and Sp + Sn = 0.

However, if one allows the illegitimate preliminary step of "

shifting" either series left or right,one can arbitrarily produce a result of any integer value you want to get. The operation of shifting (or of defining a "series" that starts with a finite number of zeros) creates a finite series that becomes the result. The amount of shifting determines the number you get, because the rest of the infinite series will still cancel.BTW, notice also that Sp + Sn = S1 from the video. So, if one rules out illegitimate cheating steps that artificially create a finite series (e.g. shifting or bracketing), the result is that

Sp + Sn = S1 = 0.