Find a divergent sequence {an} such that {a2n} converges

Let (an) be an increasing sequence of real numbers bounded from above and (bn) be a decreasing sequence. Prove that if in addition bn ≥ 1 for all n E N then the sequence (anb¹) is convergent.

B2. (a) Explain in detail what it means to say that a real sequence (an) is bounded. (b) Prove that every convergent sequence is bounded. (c) Prove that the sum of two bounded sequences is bounded. (d) Explain in detail what it means to say that a real sequence (an) diverges to ∞. (e) Suppose that the sequence (an) is bounded and the sequence (bn) diverges to . Prove that the sequence (an + bn) also diverges to ∞.

Let (an) and (bn) be sequences of real numbers such that (an) converges and (bn) diverges to +∞. Prove that (an + bn) diverges to +∞.

Is the sequence an=n3/3n-1 convergent or divergent? explain.

Solve all1) Assume that (xn) is a monotone decreasing and bounded sequence.Prove that (xn) is convergent.

Let (x₂) be an unbounded and increasing sequence of real numbers. Prove that diverges to + ∞0.

Determine whether the sequence converges or diverges n! An (п + 2)! [ Select ] In(2 + e") 3n

Find a convergent subsequence of the sequence {(-1)" (1 – ±)}, .

Find the sequence