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- True or false? Suppose (an) is Cauchy and that for every n ∈ N, the interval (−1/n,1/n) contains infinitely many terms of (an). Then (an) converges to 0Use 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)(b) Give an example to show that the product (fngn) may not converge uniformly.
- Find the interval of convergence of (3x)^n/(2n)!Apply the Ratio Test to determine convergence or divergence, or state that the Ratio Test is inconclusive.∑n=1∞enn!ρ=limn→∞|an+1an|= (Enter 'inf' for ∞.) ∑n=1∞enn! is:A. convergentB. divergentC. The Ratio Test is inconclusiveHow can I determine whether the series (-1)n+1e(-1/n+1) absolutely or conditionally converges or diverges?
- Verify if the resulting series of constant terms when x = 0 is absolutely convergent, conditionallyconvergent, or divergentSuppose that ∑ an converges absolutely. Prove that ∑ an2 converges. If we replace "converges absolutely" with "converges conditionally" does this still remain true? If yes, give a proof, if no, find a counterexample.Determine whether ∞∑n=1n√3 converges or diverges.