Euler's Constant

[Dave]

As you may or may not know, there's a very useful constant, invented by Euler which is particularly useful in applications of Number Theory and the like. It's defined to be the limit as n -> :inf: of:

 

D_n = (:sum:_{i=1 to n} 1/i) - log(n+1)

 

It's also known by the Greek letter :lcgamma: (gamma).

 

I thought I'd share this quite nice problem to show you what the value of:

 

:sum: (-1)ⁿ⁺¹/n (i.e. the infinite sum 1 - 1/2 + 1/3 - 1/4 + ...)

 

converges to. The proof goes something like this:

 

:sum: _{i = 1 to (2n-1)} (-1)ⁱ⁺¹ = 1 - 1/2 + 1/3 - 1/4 + ... + 1/(2n-1)

= 1 + 1/2 + 1/3 + ... + 1/(2n-1) - 2[1/2 + 1/4 + ... + 1/(2n-2)]

= D_2n-1 + log(2n) - D_n-1 - log(n)

= D_2n-1 - D_n-1 + log(2)

 

Now as n -> :inf:, D_2n-1 -> :lcgamma:, D_n-1 -> :lcgamma:, so

 

:sum: (-1)ⁱ⁺¹ = log(2)

 

Pretty nifty, eh?

[fafalone]

You have no idea how much I hate convergence of infinite series and proofs of it right now. Comprehensive calc2 exam with 1/3rd of it on series in 27 hours :/

[Dave]

Ah, I had to do 3 workbooks on it. Yay for comparison tests, ratio lemmas, and soforth... or not