CONVERGENCE OF p-SERIES REVISITED WITH APPLICATIONS

We construct two adjacent sequences that converge to the sum of a given convergent p-series. In case of a divergent p-series, lower and upper bounds of the (kn)th partial sum are constructed. In either case, we extend the results obtained by Hansheng and Lu (2005) to any integer k ≥ 2. Some numerical examples are given. Copyright © 2006 Hindawi Publishing Corporation. All rights reserved. Theorem 2 is the main result in this note. Lemma 1 is key to our main result. It is different from the result in [2] because it does not restrict the number of terms in the partial sum to an even one. Also, our inequalities (2) should be compared with the corresponding result in [2] for k = 2. Lemma 1. Let sn(p) be the nth partial sum of the p-series i=1(1/ip), and let k be any integer greater than 1. (a) If p> 0, then sk−1(p) − k − 1 kp k kp sn(p) < skn(p) < sk−1(p) + k kp sn(p). (1) (b) If p < 0, then k+ k kp sn−1(p) < skn(p) < 1 − 1 kp k kp sn(p). (2) Proof. Let us observe that by the definition of sn(p), we have skn(p) = 1+ 12p + ···+ 1 (kn)p k−1∑ j=0 s j kn, (3) where s j kn