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Question Number 31521 by abdo imad last updated on 09/Mar/18

study the convergence of u_n =(√(n+1)) −(√(n−1))

studytheconvergenceofun=n+1−n−1

Commented by abdo imad last updated on 12/Mar/18

we have u_n =(√n)( (√(1+(1/n))) −(√(1−(1/n))) ) but (√(1+(1/n))) ∼1 +(1/(2n)) and (√(1−(1/n))) ∼1−(1/(2n)) ⇒ u_n ∼(√n) ( (1/n)) ⇒ u_n ∼ (1/(√n)) (n→∞) so (u_n ) is convergent and lim_(n→∞) u_n =0.

wehaveun=n(1+1n−1−1n)but1+1n∼1+12nand1−1n∼1−12n⇒un∼n(1n)⇒un∼1n(n→∞)so(un)isconvergentandlimn→∞un=0.

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