Suppose an 20.If limn→∞ an+1 = L,then an 1. If 0≤ L<1,then k-1 ak converges. 2. If L> 1,then 1 ak diverges.
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- a) Suppose (an) is Cauchy and that for every k ∈ N, the interval (−1/k, 1/k) contains at least one term of (an). Can we say that (an) converges to 0? Either show that it does or give a counter-example.Prove using the ϵ−n0 definition that the sequence Xn=(9−7n)/(8−13n) converges, and find its limit.If Ean, converges and a, > 0 for all n, can anything be said about E(1/an).?
- Let a1 = 1 and an+1= √ (an+2). Prove that a nis convergent and find limn→∞ anSuppose x1 : 1/2 and xn+1 := x(n/2). Show that {xn} converges and find lim xn.Let X be a set, and define d(x,y) ={0, if x=y; 1, otherwise Show that, if a sequence (xn) is convergent under the discrete metric, then (xn) must be eventually constant.
- Suppose that ak ≥ 0 and that ak^1/3→ a as k → ∞.Prove that sigma,k from 1 to ∞,akx^k converges absolutely for all |x| < 1/a if a ≠ 0 and for all x ∈ R if a = 0Given a sequence ( xn ), prove that (xn ) converges to zero if and onlyif the sequence ( |xn| ) converges to zero.(Hint: Apply the Squeeze Theorem)Show that if a>-1 and b>a+1 , then the followingintegral is convergent.