# Logarithm and harmonic numbers

#### ZaidAlyafey

##### Well-known member
MHB Math Helper
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
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

$\chi$ $\sigma$

MHB Math Helper