Explain why if an converges, then limn→∞ an = limn→∞ an+1. With proof!
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Q: Explain why if an converges, then limn→∞ an = limn→∞ an+1.
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Explain why if an converges, then limn→∞ an = limn→∞ an+1.
With proof!
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- Use Theorem 1 to determine the limit of the sequence or state that it diverges theorem 1: If lim x--> infinity f(x) exists, then the sequence an=f(n) converges to the same limitSuppose that (sn) and (tn) are sequences so that sn = tn except for finitely many values of n. Using the definition of limit, explain why if limn → ∞ sn = s, then also limn → ∞ tn = s.Using the limit definition, prove that if {an} converges and {bn} diverges, then {an + bn} diverges.
- (a) Let (an) be a bounded (not necessarily convergent)sequence, and assume lim bn = 0. Show that lim(anbn) = 0. Why arewe not allowed to use the Algebraic Limit Theorem to prove this?Explain why if an converges, then limn→∞ an = limn→∞ an+1 . I need explanation why this theorem holds , not only proof.Suppose that {xn} is a sequence of real numbers satisfying lim (as n→∞) xn = 1. prove that lim (as n→∞) (1 + 2xn) = 3.
- Let bn = an+1. Use the limit definition to prove that if {an} converges, then {bn} also converges and lim n→∞ an = limn→∞ bnSuppose that {xn} is a sequence of real numbers satisfying lim (as n→∞) xn = 1. prove that lim (as n→∞) (1 + 2xn) = 3. prove this don't calculate limit.Prove that limx → c f(x) = ∞ if and only if for every sequence an that is in the domain of f and converges to c (but never equal to c), we have limn → ∞ f(an) = ∞. (That is, prove a version of Relating Sequences to Functions for infinite limits. Make sure you handle both directions of the if and only if!)
