Advanced Calculus fi..

Advanced Calculus, Fifth Editionwhich, when added, become the convergent seriesEXAMPLE 13xz2(,oiny.Here the root test is easily applicable:m=-1log n 'so that the root approaches 0 as limit, and the series converges.EXAMPLE 14 Em2 (*). . Again the root test proves convergence:PROBLEMSnn-*m 1 + n2lim = lim - - 0.n+m1. Prove divergence by the nth term test:a) CEl sin (q)2"b) CE, 72. Prove convergence by the comparison test:00 1a) Cn=2 b) CEl $p3. Prove divergence by the comparison test:n+5 1a) CZl n2 - 3n - 5 b) CZ24. Prove convergence by the integral test: c -c -.-00 1 00 18) En=, b) Cn=25. Prove divergence by the integral test:I-w;6. Determine convergence or divergence by the ratio test:a) CZl 9zn + 1b) CEl7. Prove convergence by the alternating series test:tr\ I'i 7 bn:a) CE, gb) CE2 (- 1)" log nn8. Prove convergence by the root test:m 1;;ia) En=,b) CEI (A) n29. Prove convergence by showing that the nth partial sum converges:1 nfl na) CZI (n +2)(n + 1) = CZl (m- m)m 1-nb)C=l - CZ1(* - $110. Prove the validity of the integral test (Theorem 14) for divergence.11. Prove the validity of the alternating series test (Theorem 18) by applying the Cauchycriterion (Theorem 9).