convergence of series.

**jefferson_lc** · November 25th 2010, 06:57 PM

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.

**chisigma** · November 25th 2010, 07:39 PM

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

$\chi$ $\sigma$

**tonio** · November 26th 2010, 04:21 AM

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

Thread: convergence of series.

LinkBack

Thread Tools

Display

convergence of series.

Similar Math Help Forum Discussions

Show that a series converges absolutely given convergence of two related series

radius of convergence and interval of convergence of the series

Proove that convergence of one series implies the convergence of another series

Proove that convergence of one series implies the convergence of another series

series convergence and radius of convergence

Search Tags