TY - JOUR

T1 - Two Convergence Theorems and an Extension of the Ratio Test for a Series

AU - Wu, Zili

N1 - Publisher Copyright:
© 2019, © Mathematical Association of America.

PY - 2019/5/27

Y1 - 2019/5/27

N2 - Summary: Given a decreasing sequence (Formula presented.) of nonnegative terms, we prove that there exist subsequences (Formula presented.) of {n} such that (Formula presented.) and (Formula presented.) have the same convergence. In particular, (Formula presented.) converges if and only if (Formula presented.) converges for any natural number q > 1. Using this result, the ratio test is also extended. One example further shows that the new extended ratio test is applicable to discussing the convergence of a p-series for which the ratio test fails.

AB - Summary: Given a decreasing sequence (Formula presented.) of nonnegative terms, we prove that there exist subsequences (Formula presented.) of {n} such that (Formula presented.) and (Formula presented.) have the same convergence. In particular, (Formula presented.) converges if and only if (Formula presented.) converges for any natural number q > 1. Using this result, the ratio test is also extended. One example further shows that the new extended ratio test is applicable to discussing the convergence of a p-series for which the ratio test fails.

KW - MSC: Primary 40A05

UR - http://www.scopus.com/inward/record.url?scp=85074966159&partnerID=8YFLogxK

U2 - 10.1080/0025570X.2019.1560841

DO - 10.1080/0025570X.2019.1560841

M3 - Article

AN - SCOPUS:85074966159

SN - 0025-570X

VL - 92

SP - 222

EP - 227

JO - Mathematics Magazine

JF - Mathematics Magazine

IS - 3

ER -