Question about series on knopp book The book of K. Knopp, Theory and Application of infinite series , 1990
on page 312, exercise 135, its written
"For every fixed $\rho$ in $0<\rho<1$, the expression 
$\sum_{\gamma=1}^{n}[\frac{1}{\gamma^{1-\rho}}-\frac{n^{\rho}}{\rho}]$
has a definite limit $\gamma_{\rho}$ when $n\rightarrow+\infty$
I trully belive that it have a problem (=misprint) with the position of the brackets.
I did'nt find the errata of the book. So i would like any help. 
 A: This appears to be analogous to the definition of Euler's constant:
$$\lim_{n \to \infty} \left[\sum_{k=1}^n \frac1{k} - \ln n\right] = \gamma.$$
The left bracket is probably misplaced, as you suspect, and the expression in Knopp should be written as
$$\lim_{n \to \infty} \left[\sum_{k=1}^n \frac1{k^{1-\rho}} - \frac{n^{\rho}}{\rho}\right] = \gamma_\rho.$$
With this interpretation, existence of the limit follows by showing that the sequence is bounded and monotonically decreasing.
Note that
$$\sum_{k=1}^{n-1} \frac1{(k+1)^{1-\rho}} < \sum_{k=1}^{n-1} \int_k^{k+1} \frac{dx}{x^{1-\rho}}< \sum_{k=1}^{n-1} \frac1{k^{1-\rho}}.$$
Hence,
$$\sum_{k=1}^{n} \frac1{k^{1-\rho}}-1 < \int_1^{n} \frac{dx}{x^{1-\rho}}= \frac{n^\rho}{\rho}- \frac{1}{\rho}< \sum_{k=1}^{n} \frac1{k^{1-\rho}}- \frac{1}{n^{1-\rho}},$$
and
$$-\frac{1}{\rho}< \frac{1}{n^{1-\rho}}-\frac{1}{\rho}<\sum_{k=1}^{n} \frac1{k^{1-\rho}}- \frac{n^\rho}{\rho}<1- \frac{1}{\rho}.$$
Therefore, the sequence is bounded. 
I'll leave it to you to show that the sequence is monotone.
