In the current issue of the Mathematical Gazette there is an article on slowly convergent series. There the tortoise challenges Achilles to sum the series:

$\displaystyle K=\sum_{n=0}^{\infty} (-1)^n (n+1)^{-\frac{n+2}{n+1}}$

to four significant figures.

Now far as I can see; this is an alternating series with terms of strictly decreasing absolute value which go to zero as $\displaystyle n$ goes to infinity and so it converges to a limit between $\displaystyle 1$ and $\displaystyle 1-2^{-3/2}$ (between $\displaystyle 1$ and $\displaystyle 0.64$).

Also numerical experiments reassuringly suggest that the limit is $\displaystyle \approx 0.78$

However the paper seems to think the sum $\displaystyle K \approx 1.32..$. Which leaves me puzzled, what have I done wrong, or what could be the misprint in the expression for the series?

CB