# convergence of series.

• Nov 25th 2010, 07:57 PM
jefferson_lc
convergence of series.
How does one prove that the series

1 + 1/3 - 1/2 + 1/5 + 1/7 - 1/4 + 1/9 + 1/11 - 1/6 + ...

converge?

PS.: This series is a rearrangement of the alternating harmonic series in which two positive terms are always followed by one negative.
• Nov 25th 2010, 08:39 PM
chisigma
The series converges... but not as $\displaystyle 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + ... = \ln 2$ because that is a weakly convergent series...

Kind regards

$\chi$ $\sigma$
• Nov 26th 2010, 05:21 AM
tonio
Quote:

Originally Posted by jefferson_lc
How does one prove that the series

1 + 1/3 - 1/2 + 1/5 + 1/7 - 1/4 + 1/9 + 1/11 - 1/6 + ...

converge?

PS.: This series is a rearrangement of the alternating harmonic series in which two positive terms are always followed by one negative.

Its limit is $\displaystyle{\frac{3}{2}\ln 2}$ .

The proof appears in Bonar-Khouri's "Real Infinite Series", in 3.10.

Tonio