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- 4.). Find the interval of convergence. (Enter your answer using interval notation.) ∞ n = 0 xn n6 + 3 please show step by step clearlythe nth Term Divergence Test (Theorem 3) to prove that the following series diverge.1). Apply the ratio test to the series.∞n!6n3 n=1 Find the limit lim n→∞ an+1an . Use your result to determine the convergence of the series.absolutely convergentdivergent inconclusive please show step by step .
- 2. Consider the series partial sum of infinity to k = 2 of (k^4 + 2k^2 + k −3)^(1/2)/(k^3 + 2k^2 + 6) .(a) Explain why the Limit Comparison Test is a good choice for determining convergence/divergence of theseries.(b) Use the Limit Comparison Test to determine whether the converges or diverges. Make sure to carefully apply the test and to explain your choice of a comparison series.1) MacLaurin’s Formula that includes the 7-th Taylor reminder,2) the radius of convergence,3) the interval of convergenceonly 10,12,16: test the series of convergence/divergence.
- 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)Determine the radius of convergence of the series 1 + x/2 + x2/3 + x3/4 +...1)Determine whether the Fourier series of the following functions converge uniformly or not. A) f(x) = ex, - 1 < x < 1 ***Solve using power and series (Fourier)(please)** answer: a) The periodic function is not continuous at ± π etc., so the convergence cannot be uniform.
- F(x) = (2n − 1)!/(2n + 1)! Does the series converge or diverge, if it converges find the limit. Show all necessary work and state any necessary tests or theorems used.1). Apply the ratio test to the series. ∞ n! 3n3 n=1 Find the limit lim n→∞ an+1 an . Use your result to determine the convergence of the series. absolutely convergent divergent inconclusive please show step by step clearlyuppose that the sn satisfies both limn→∞ s2n = 3 and limn→∞ s2n+1 = 3. (That is, the sequence given by the even terms of sn and that given by the odd terms of sn both converge to 3.) Show that also limn→∞ sn = 3.ii. Give an example of a sequence where the sequences given by the even and by the odd terms both converge, but where the entire sequence does not converge.