View source for Dedekind criterion (convergence of series)

The series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030520/d0305201.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030520/d0305202.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030520/d0305203.png" /> are complex numbers, converges if the series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030520/d0305204.png" /> converges absolutely, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030520/d0305205.png" />, and if the partial sums of the series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030520/d0305206.png" /> are bounded.



====Comments====
The proof is based on the formula of summation by parts: Put <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030520/d0305207.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030520/d0305208.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030520/d0305209.png" />. Then for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030520/d03052010.png" /> one has

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030520/d03052011.png" /></td> </tr></table>

A related convergence criterion is the [[Dirichlet criterion (convergence of series)|Dirichlet criterion (convergence of series)]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Rudin,   "Principles of mathematical analysis" , McGraw-Hill  (1953)</TD></TR></table>