Geometric series and its derivatives

[thepatient]

Homework Statement

I was browsing online and stumbled upon someone's explanation as to why 1 -2 +3 -4 + 5... towards infinity= 1/4. His explanation didn't make sense to me. He starts with a geometric series, takes a derivative, and plugs in for x = -1, and gets a finite value of 1 -2 + 3 - 4 +... = 1/4. It doesn't make sense to me that he would plug in for x = 1, since the series equals 1/(1-x) only for |x|< 1 and converges when |x| <1. So I wanted to know what you guys think.

Homework Equations

∑₀^∞xⁿ = 1/(1-x), |x| < 1

The Attempt at a Solution

This is how the work is shown. He wants to show that 1-2+3-4+5-...=1/4. He begins with:

∑₀^∞xⁿ = 1/(1-x), |x| < 1
1 + x + x² + x³ +... = 1/(1-x)

Takes a derivative of both sides:

1 + 2x + 3x² + 4x³ + ... = 1/(1-x)²

plugs in for x = -1

1 -2 +3 -4 + 5... = 1/4

Are these operations valid?

 

[andrewkirk]

Are these operations valid?

No, because the requirement for the infinite sum to converge was |x|<1, which is not obeyed by setting x=-1.

 

[thepatient]

That is what I thought as well. But apparently, this result is used to prove that 1 + 2 + 3 + 4 +5... = -1/12, and apparently that result is used in physics like String Theory.