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- Use the Cauchy Condensation test to prove that ∑ n = 2 to ∞ 1/( n (ln(n))^ p)) converges if p > 1 and diverges if p ≤ 1. (Make sure you verify that the hypothesis of the Cauchy Condensation test are met)Find the interval of convergence of ∞∑n=0 ((-1)n (x+1)n )/ (2n).Find interval of convergence Summation from n =1 to infinity of ((n(x+4))^n)/3^n
- only 10,12,16: test the series of convergence/divergence.Apply the Cauchy’s n-th Root Test tothe given series. Possible answers are «convergent», «divergent», or «Cauchy’sn-th Root Test gives no information».8.7.4. Find the Maclaurin series (i.e., Taylor series about c = 0) and its interval of convergence.
- without using the Cauchy Completeness Theorem, show that if (sn) and (tn) are both Cauchy sequences, then (sn ⋅ tn) is also Cauchy.Determine the radius of convergence of the series 1 + x/2 + x2/3 + x3/4 +...Use limit comparison test to determine convergence or divergence of Sum from n=1 to ♾ of (3n+1)/(n^4+2n^2+7)
- Let (sn) be a sequence with lim sup sn = 10 and lim inf sn = 5.(a) Show that (sn) is bounded.(b) Construct a sequence (sn) with lim sup sn = 10 and lim inf sn = 5 and withinfinitely many subsequential limits. (It is redundant to say this but just toemphasize: you need to prove that your constructed sequence has infinitelymany subsequential limits and that lim sup sn = 10 and lim inf sn = 5.(a) Carefully determine the convergence of the seriesIf a power series ∞ n=0 anxn converges absolutely at a pointx0, then it converges uniformly on the closed interval [−c, c], where c = |x0|.