Why is there a 'missing' $1$ in the Euler–Mascheroni constant? It is easy to show that:
$$
\sum_{k=1}^n \frac{1}{k} > \ln(n+1),
$$
but the Euler–Mascheroni constant is defined as:
$$
\gamma = \lim_{n \to \infty} \left(  \sum_{k=1}^n \frac{1}{k} - \ln(n) \right).
$$
My question is, why was $\gamma$ defined using $\ln(n)$ and not  $\ln(n+1$)? 
Are the two definitions identical, or does it simply turn out to be more convenient for other applications to define $\gamma$ using $\ln(n)$?

$$ \left(  \sum_{k=1}^{n} \dfrac{1}{k} - \ln(n+1) \right) - \left(  \sum_{k=1}^{n} \dfrac{1}{k} - \ln(n) \right)=\ln(n)-\ln(n+1)=\ln\left(\frac{n}{n+1}\right)$$
And 
$$\lim_n \ln\left(\frac{n}{n+1}\right)=\ln 1=0$$
