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 -