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

Printable View

- May 27th 2009, 05:45 AMNaNaReal 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 AMmr fantastic
- May 27th 2009, 06:00 AMchisigma
Starting from the geometric series...

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

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

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

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

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

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$ - May 27th 2009, 06:02 AMNaNa
So easy thanks a lot!!!