Construct two divergent sequences (x′n)n≥1 and (y′n)n≥1 such that (x′ny′n)n≥1 converges
Question
Construct two divergent sequences (x′n)n≥1 and (y′n)n≥1 such that (x′ny′n)n≥1 converges
Solution
Sure, here are two divergent sequences whose product converges:
-
Let's define the sequence (x'n)n≥1 as x'n = n. This sequence is divergent because it goes to infinity as n increases.
-
Now, let's define the sequence (y'n)n≥1 as y'n = 1/n. This sequence is also divergent because it goes to zero as n increases.
However, if we take the product of these two sequences, we get (x'n * y'n)n≥1 = n * 1/n = 1 for all n. This sequence is not only convergent, but it is a constant sequence, which means it stays the same for all n.
Similar Questions
Construct two divergent sequences (xn)n≥1 and (yn)n≥1 such that (xn + yn)n≥1converges
Let (xn)n≥1 and (yn)n≥1 be bounded sequence. Prove thatlim infn→∞ (xn + yn) ≥ lim infn→∞ xn + lim infn→∞ yn
Define (xn)n≥1 by x1 := 0 and xn+1 := (xn + 3)/4 for n ≥ 1. Prove that the sequence (xn)n≥1 isconvergent and calculate the limit limn→∞ xn
All the following sequences converge to 1 except ..........*1 pointv_n={ n/(2n+1) }u_n={ 2n/(2n+1) }w_n={ (1+n)/(7+n) }z_n={ (n+3)/(n+4) }
Two chromosomes that look alike and have the same sequence of genes in the same positions are a(n) ______ pair.
Upgrade your grade with Knowee
Get personalized homework help. Review tough concepts in more detail, or go deeper into your topic by exploring other relevant questions.