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Logarithm and harmonic numbers

ZaidAlyafey

Well-known member
MHB Math Helper
Jan 17, 2013
1,667
I need to prove that

\(\displaystyle H_n = \ln n + \gamma + \epsilon_n \)

Using that

\(\displaystyle \lim_{n \to \infty} H_n - \ln n = \gamma \)

we conclude that

\(\displaystyle \forall \, \epsilon > 0 \,\,\,\, \exists k \,\,\,\, \) such that \(\displaystyle \,\,\, \forall k \geq n \,\,\, \) the following holds

\(\displaystyle |H_n - \ln n -\gamma | < \epsilon \)

\(\displaystyle H_n < \ln n +\gamma +\epsilon \)

I think I used the wrong approach , didn't I ?
 

chisigma

Well-known member
Feb 13, 2012
1,704
Re: logarithm and harmonic numbers

I need to prove that

\(\displaystyle H_n = \ln n + \gamma + \epsilon_n \)

Using that

\(\displaystyle \lim_{n \to \infty} H_n - \ln n = \gamma \)

we conclude that

\(\displaystyle \forall \, \epsilon > 0 \,\,\,\, \exists k \,\,\,\, \) such that \(\displaystyle \,\,\, \forall k \geq n \,\,\, \) the following holds

\(\displaystyle |H_n - \ln n -\gamma | < \epsilon \)

\(\displaystyle H_n < \ln n +\gamma +\epsilon \)

I think I used the wrong approach , didn't I ?
http://mathhelpboards.com/discrete-...ation-tutorial-draft-part-i-426.html#post2494

Kind regards

$\chi$ $\sigma$
 

ZaidAlyafey

Well-known member
MHB Math Helper
Jan 17, 2013
1,667