Use Limit Comparison Test to find the condition on p that determine the convergence or divergence of the series 1 1+ 2P 1 1 3P NP n=1 prove that Ean diverges if lim nan = 1. n-00
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- Using the limit definition, prove that if {an} converges and {bn} diverges, then {an + bn} diverges.Test for convergence/divergence and state the theorems used.Assume fn → f on a set A. Theorem 6.2.6 (Continuous Limit Theorem).is an example of a typical type of question which asks whether a trait possessed by each fn is inherited by the limit function. Provide an example to show that all of the following propositions are false if the convergence is only assumed to be pointwise on A. Then go back and decide which are true under the stronger hypothesis of uniform convergence. (d) If each fn has fewer than M discontinuities (where M ∈ N is fixed), thenf has fewer than M discontinuities.
- 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 limitProve formally whether the given sequence has a limit, using the Cauchy criterium. If the sequence is properly divergent prove it formally too.Use Monotone Convergence Theorem to prove that (Sn) converges and find its limit.
- Let fn be a bounded sequence of functions uniformly convergetn to f. Prove that f is bounded as well. Is the claim true if we replace the assumption of uniform convergence with pintwise convergence.Use limit comparison test to determine convergence or divergence of Sum from n=1 to ♾ of (3n+1)/(n^4+2n^2+7). Let gn = nχ[1/n,2/n] and g = 0. Show thatZ 20g 6= limn→∞ Z 20gn.Does the sequence (gn) converge uniformly to g? Does the Monotone Convergence Theoremapply? Does Fatou’s Lemma apply?