In mathematics, the limit comparison test (LCT) (in contrast with the related direct comparison test) is a method of testing for the convergence of an infinite series.
Statement Suppose that we have two series Σ n a n {\displaystyle \Sigma _{n}a_{n}} and Σ n b n {\displaystyle \Sigma _{n}b_{n}} with a n ≥ 0 , b n > 0 {\displaystyle a_{n}\geq 0,b_{n}>0} for all n {\displaystyle n} . Then if lim n → ∞ a n b n = c {\displaystyle \lim _{n\to \infty }{\frac {a_{n}}{b_{n}}}=c} with 0 < c < ∞ {\displaystyle 0<c<\infty } , then either both series converge or both series diverge.
Proof Because lim n → ∞ a n b n = c {\displaystyle \lim _{n\to \infty }{\frac {a_{n}}{b_{n}}}=c} we know that for every ε > 0 {\displaystyle \varepsilon >0} there is a positive integer n 0 {\displaystyle n_{0}} such that for all n ≥ n 0 {\displaystyle n\geq n_{0}} we have that | a n b n − c | < ε {\displaystyle \left|{\frac {a_{n}}{b_{n}}}-c\right|<\varepsilon } , or equivalently
− ε < a n b n − c < ε {\displaystyle -\varepsilon <{\frac {a_{n}}{b_{n}}}-c<\varepsilon }
c − ε < a n b n < c + ε {\displaystyle c-\varepsilon <{\frac {a_{n}}{b_{n}}}<c+\varepsilon }
( c − ε ) b n < a n < ( c + ε ) b n {\displaystyle (c-\varepsilon )b_{n}<a_{n}<(c+\varepsilon )b_{n}}
As c > 0 {\displaystyle c>0} we can choose ε {\displaystyle \varepsilon } to be sufficiently small such that c − ε {\displaystyle c-\varepsilon } is positive. So b n < 1 c − ε a n {\displaystyle b_{n}<{\frac {1}{c-\varepsilon }}a_{n}} and by the direct comparison test, if ∑ n a n {\displaystyle \sum _{n}a_{n}} converges then so does ∑ n b n {\displaystyle \sum _{n}b_{n}} . Similarly a n < ( c + ε ) b n {\displaystyle a_{n}<(c+\varepsilon )b_{n}} , so if ∑ n a n {\displaystyle \sum _{n}a_{n}} diverges, again by the direct comparison test, so does ∑ n b n {\displaystyle \sum _{n}b_{n}} . That is, both series converge or both series diverge.
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