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Converges to l. example: prove directly from the definition that the sequence n+3 a, nen an 2n+1 is a cauchy sequence. solution: let s> 0 be given. let n be a p
Converges to L. Example: Prove directly from the definition that the sequence n+3 a, neN an 2n+1 is a Cauchy sequence. Solution: Let s> 0 be given. Let N be a positive integer to be chosen. Suppose that n, m 2 N. We have m+3 m-n a,--3 2n+1 2m+1 (2n+1)(2m+1) (2n+1)+ (2m+1) -2 (2n+1)(2m+1) 2m+2n (2n+ 1)(2m+1) 1 1 2 2m+1 2n+1 (2n +1)(2m +1) 1 2m+1 2n+1 1
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Priyadarshan Choursiya![best-answer]()
you note that every Cauchy sequence is convergent and vice versa. so you should do directly by find limit of an. you see that limit n tends to infinity of an gives you 0.5 therefore seq. (an) is convergent so cauchy. for batter understanding of Cauchy sequence visit https://youtu.be/wIIxmjFXj2A https://youtu.be/B-7cUVXSZeI