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Q: Determine the following limits:
(a) lim((3√n)1/2n), (b) lim((n+1)1/ln(n+1)).
(a) lim((3√n)1/2n), (b) lim((n+1)1/ln(n+1)).
Q. Let yn=√(n+1)-√n for nϵN. Show that (yn) and (√nyn) converge. Find their limits.
Q. Explain why the result in question. (show that if X and Y are sequences such that X and X+Y are convergent, then Y is convergent) before theorem 3.2.4 of book real analysis, 3rd edition, by Robert G Bartle cannot be used to evaluate the limit of the sequence ((1+1/n)n).
Q: If (bn) is a bounded sequences and lim(an)=0, show that lim(anbn)=0 (explain why the theorem 3.2.3 from book real analysis 3rd edition, by Robert G Bartle can not be used)
Q: Find the limit of the following sequences:
(a) lim2+(1/n2), (b)lim(-1)n/(n+2)
(c)lim((√n-1)/(√n+1))
(d)lim((n+1)/(n(√n)))
(a) lim2+(1/n2), (b)lim(-1)n/(n+2)
(c)lim((√n-1)/(√n+1))
(d)lim((n+1)/(n(√n)))
Q: Show that the following sequences are not convergent.
(a) (2n), (b)((-1)nn2)
(a) (2n), (b)((-1)nn2)
Q. show that if X and Y are sequences such that X converges to x≠ 0 and XY converges, then Y converges.
Q. show that if X and Y are sequences such that X and X+Y are convergent, then Y is convergent.
Q: Give an example of two divergence sequences X and Y such that:
(a) their sum X+Y converges, (b) their product XY converges
(a) their sum X+Y converges, (b) their product XY converges
Q. For xn given by the following formulas, establish either the convergence or divergence of the sequence X=(xn)
(a) xn= n/(n+1), (b) xn=(〖(-1)〗^n n)/(n+1) ,
(c) xn=n2/(n+1), (d)xn=(2n2+3)/(n2+1)
(a) xn= n/(n+1), (b) xn=(〖(-1)〗^n n)/(n+1) ,
(c) xn=n2/(n+1), (d)xn=(2n2+3)/(n2+1)
