Assume that each of the two limits Lo = lim n n>1 if, the other series...

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Assume that each of the two limits Lo = lim n→∞ n>1 if, the other series converges. |a2n| |an| and L₁ lim n→∞ |a2n+1| |an| exists or is infinite, that is, Lo, L₁ € RU{∞}. R1: Pick a natural number N≥ 1. For k ≥ 0, define Ak = = (|a₂kN| + |a₂kN+1]) + (|a₂kN+2] + [a2kN+3|) + · + (|a2k+1N_2| + |a2k+¹N−1|) Write an in terms of Σ Ak, and show that one series converges if, and only k>0 R2: Suppose M = max{Lo, L₁} is finite. Show that we can choose a natural number N and a real number r≥ M such that Ak+1 ≤ 2r Ak for all k ≥ 0. 1 R3: Show that if M max{Lo, L₁} < then the series an converges absolutely. n>1 R4: Let m = min{Lo, L₁}. Show that we can choose a natural number N and a real number r ≤ m such that Ak+1 ≥ 2r Ak for all k ≥ 0. 1 R5: Show that if m= min{Lo, L₁} > then the series Σ 2 n>1 an diverges. Assume that each of the two limits Lo = lim n→∞ n>1 if, the other series converges. |a2n| |an| and L₁ lim n→∞ |a2n+1| |an| exists or is infinite, that is, Lo, L₁ € RU{∞}. R1: Pick a natural number N≥ 1. For k ≥ 0, define Ak = = (|a₂kN| + |a₂kN+1]) + (|a₂kN+2] + [a2kN+3|) + · + (|a2k+1N_2| + |a2k+¹N−1|) Write an in terms of Σ Ak, and show that one series converges if, and only k>0 R2: Suppose M = max{Lo, L₁} is finite. Show that we can choose a natural number N and a real number r≥ M such that Ak+1 ≤ 2r Ak for all k ≥ 0. 1 R3: Show that if M max{Lo, L₁} < then the series an converges absolutely. n>1 R4: Let m = min{Lo, L₁}. Show that we can choose a natural number N and a real number r ≤ m such that Ak+1 ≥ 2r Ak for all k ≥ 0. 1 R5: Show that if m= min{Lo, L₁} > then the series Σ 2 n>1 an diverges.

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Proof Assume that each of the two limits Lo lim 192n Tanl L lim logn 1 lanl n700 n exist... View the full answer