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Solved in 2 steps
- US THE LIMIT CAMPARISON TESTS TO DETERMINE THECONVERGENCES OR DIVERGECNES OF THE SERIES,without using the Cauchy Completeness Theorem, show that if (sn) and (tn) are both Cauchy sequences, then (sn ⋅ tn) is also Cauchy.With the help of partial fraction or otherwise, find the series representation of a) about z = 0 for the region 2 < |z| < 3, b) about z = 2 for the region 1 < |z − 2|. (Hint: let w = z − 2.) In each case, write explicitly some non-vanishing terms to illustrate the series.
- does this geometric seris converge to... 4/5 4/3 2/5 2/3Does the series 1+1/8+1/27+1/64+1/125+...(reciprocals of the perfect cudes) converge or diverge? ExplainApply the De Almbert’s Ratio Test tothe given series. Possible answers are «convergent», «divergent», or «DeAlmbert’s Ratio Test gives no information».
- 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)Suppose that the radius of convergence of the power series cn xn is R. What is the radius of convergence of the power series cn x7n ?Does a(n)= n^2sin(3/n^2) converge? If so, to where?