An alternating sum

[Heptadecagon]

Hello, I have recently encountered the sum ln(1)/1 - ln(2)/2 + ln(3)/3 - ln(4)/4 + ln(5)/5 - ln(6)/6 ..... = sum n=1 to infinity -(-1)^n*ln(n)/n
It is clear it converges, but does anyone know the answer and have a proof? Thank you!

[cyanyoshi]

Just because a series converges does not mean that there is necessarily a simple, pretty expression that it simplifies to. After a bit of hunting around on Wikipedia, this series may equal d/ds Li_s(-1) evaluated at s = 1, where Li_s(z) is the polylogarithm function. Whether this is in any way more elegant than the original expression is up for debate, though. There are probably at least a few other ways to write this using existing functions and operators.

Spoiler:

Li_s(z) ≡ ∑^∞_k=1 z^k/k^s
Li_s(-1) = ∑^∞_k=1 (-1)^k/k^s
d/ds Li_s(-1) = ∑^∞_k=1 d/ds (-1)^k/k^s (not 100% sure if differentiating term-by-term is ok)
d/ds Li_s(-1) = ∑^∞_k=1 d/ds (-1)^k k^-s
d/ds Li_s(-1) = ∑^∞_k=1 -(-1)^k ln(k) k^-s
d/ds Li_s(-1)|_s=1 = ∑^∞_k=1 -(-1)^k ln(k) / k

[Heptadecagon]

Thank you very much, while that probably won't help me find a closed form that I would like, it is very interesting, as Lis(-1) is related to the Dirichlet eta function (and is in fact it's opposite) , and right above references on this page: http://en.wikipedia.org/wiki/Dirichlet_eta_function is a formula for the derivative that I want... Which unfortunately is undefined at s=1 , and while the limit should exist I have no idea on how to evaluate it... I believe this sum has a closed form with natural logarithms and the Euler-Mascheroni constant because of some integrals I've been fooling around with, but I'm not quite sure yet. Thank you for your help!

[PM 2Ring]

If it's any help, I get a value of -0.159868(9), using brute-force summation to 10000 terms, followed by Richardson approximation.

To 25000 terms:
-0.1598688994744

although I suspect that the last 4 or 5 digits are garbage.

[Heptadecagon]

Thank you, although I probably do not need too many digits, just enough to make me feel confident with my answer, once I get it. Wolfram-Alpha spits out hundreds of digits, although I'm sure it messes up quite quickly, who knows. All I know is that I have plenty of digits, thanks for the support! I know I will solve it, just one more crazy integral to go...

[eta oin shrdlu]

I believe this sum has a closed form with natural logarithms and the Euler-Mascheroni constant because of some integrals I've been fooling around with, but I'm not quite sure yet. Thank you for your help!

You are right. Here's one approach:

Spoiler:

Consider the finite sum (to avoid convergence issues) and split it into two nonalternating series:

∑^2N_n=1 (-1)^n-1 ln n/n
= ∑^2N_n=1 ln n/n - 2 ∑^N_n=1 ln(2n)/(2n)
= ∑^2N_n=N+1 ln n/n - ln 2 ∑^N_n=1 1/n

It's easy to see that the second series is roughly ln 2 ln N, and
lim_N→∞ ln 2 ∑^N_n=1 1/n - ln 2 ln N = γ ln 2 .

You can turn the first series into Riemann sums,
∑^2N_n=N+1 (ln N + ln(n/N))/n
= ln N ∑^2N_n=N+1 1/(n/N) 1/N + ∑^2N_n=N+1 ln(n/N)/(n/N) 1/N .

The first of these series approaches
ln N ∫₁² dx/x = ln N ln 2
which cancels the ln 2 ln N divergence in the limit above.

The second of these series approaches
∫₁² ln x/x dx = (ln 2)²/2 .

You can be more rigorous about taking the N→∞ limit to keep everything convergent; this gives the closed form
(ln 2)²/2 - γ ln 2 .

This sum was recently posed at Les Reid's site; I don't know its earlier history.

[Heptadecagon]

Cool, thanks for the help, it's interesting to see such different methods. I've already seen the -ln(2)*γ term, but not the other quite yet. I understand the proof you posted except to
''The second of these series approaches
∫12 ln x/x dx = (ln 2)2/2 . '' , but I have just woke up and haven't got out any paper yet. Thanks for all the help, and even if someone has another method of solving it it would be interesting to see, and it would also be neat to know the history of this sum. Thanks!

[PM 2Ring]

Spoiler:

this gives the closed form
(ln 2)²/2 - γ ln 2 .

Ah. That's approximately equal to

Spoiler:

-0.
15986 89037 42430 97175 69478 70324 91657 04962 22023 75645 87426 70824 52963 96570 02184 02900 46595 55034 03204 61882 94631 63497 61658 09595 68086 66967 72963 15393 02307 37835 50325 63252 06746 52469 17616 61984 85991 58220 76417 90373 21995 25828 12903 38761 56808 89208 39793 25153 43900 76645 14516 53255 13764 04909 01383 30873 42453 20448 79971 82138 70575 20589 08712 20903 33414 76723 13593 90886 78402 26598 94944 98785 08283 73343 31108 58069 02559 36075 82095 55055 51644 72474 39148 73149 85186 96793 60964 74081 50918 75189 22740 09964 40326 05964 26344 41972 77684 71219 06241 10813