Logarithm and harmonic numbers

[alyafey22]

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]

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-mathematics-set-theory-logic-15/difference-equation-tutorial-draft-part-i-426.html#post2494

Kind regards

$\chi$ $\sigma$

 

[alyafey22]

Re: logarithm and harmonic numbers

http://mathhelpboards.com/discrete-mathematics-set-theory-logic-15/difference-equation-tutorial-draft-part-i-426.html#post2494

Kind regards

$\chi$ $\sigma$

My friend this is amazing , I must have time to read that , keep it up .