# Real Analysis

• May 27th 2009, 05:45 AM
NaNa
Real Analysis
is it true that 1-1/2+1/3-1/4=....=ln2??

how i solve that question i have to use the harmonic series!?

Thanks
Nana
• May 27th 2009, 05:53 AM
mr fantastic
Quote:

Originally Posted by NaNa
is it true that 1-1/2+1/3-1/4=....=ln2??

how i solve that question i have to use the harmonic series!?

Thanks
Nana

Substitute x = 1 into the Maclaurin series for ln(x + 1).
• May 27th 2009, 06:00 AM
chisigma
Starting from the geometric series...

$\frac{1}{1+x} = 1 - x + x^{2} - x^{3} + ...$ , $|x|<1$ (1)

... and integrating term by term we obtain...

$\ln (1+x) = x - \frac{x^{2}}{2} + \frac{x^{3}}{3} - \frac{x^{4}}{4} + ...$ (2)

Setting in (2) x=1 we have...

$\ln 2 = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + ...$ (3)

Kind regards

$\chi$ $\sigma$
• May 27th 2009, 06:02 AM
NaNa
So easy thanks a lot!!!