LIMIT COMPARISON TEST: Let n=1∑∞​an​ and n=1∑∞​bn​ be infinite series with positive terms. (1) If n→∞lim​bn​an​​=c=0, t

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Limit Comparison Test Let N 1 An And N 1 Bn Be Infinite Series With Positive Terms 1 If N Lim Bn An C 0 T 1 (122.18 KiB) Viewed 37 times

LIMIT COMPARISON TEST: Let n=1∑∞​an​ and n=1∑∞​bn​ be infinite series with positive terms. (1) If n→∞lim​bn​an​​=c=0, then either both series converge or both series diverge. (2) If n→∞lim​bn​an​​=0 and n=1∑∞​bn​ converges, then n=1∑∞​an​ converges. (3) If n→∞lim​bn​an​​=∞ and n=1∑∞​bn​ diverges, then n=1∑∞​an​ diverges. Our objective is to determine the behavior of given series. We generally represent the given series by n=1∑∞​an​. As with the Direct Comparison Test, we must come up with a new series to which to compare the given series, and we must know how this new series behaves. We refer to this new series as the contrived series, and we represent it by n=1∑∞​bn​. Once we have the given series and the contrived series, we can apply the Limit Comparison Test.
In problems 1 (a-d) a series is given. Come up with the contrived series (the series to which to compare the given series), and apply the Limit Comparison Test to determine whether the given series converges or diverges. (a) n=1∑∞​n2−11​ Converges (b) n=1∑∞​2n−11​ Converges (c) n=1∑∞​n2+1​1​ Diverges (d) n=1∑∞​3n+12n​ Converges