In the first part of this bachelor thesis, some basic concepts related to numerical series as well as the basic definition of numerical, geometric, harmonic, hyper-harmonic and alternating series are presented. Also, some necessary theorems are listed and a limit comparison test, ratio test and integral test for identifying convergence of some numerical series are presented. In the second part, the focus is on evaluating the mistake when estimating the sum of a series with positive terms. We show that the limit comparison test, ratio test and integral test can be used to find both lower and upper estimates for the sum of a series.