Problem 5. Kummer's Test: let Σan be a series of positive terms and suppose {pn} is a sequence of positive numbers such that p = lim (pn An n→X An+1 - Pn+1) exists. (i) If p>0 then Σ an converges. 1 (ii) If p < 0 and Σ = ∞ then an diverges. Pn (iii) If p=0, the test is inconclusive. (More general versions with necessary and sufficient conditions can be found online, see also posted article). (a) Show that the root test (Theorem 7.8 in the text) is a special case of Kummer's test with pn = 1. (b) Example from Part 3 of the proof of Theorem 7.9 in the text: let a2 = 2 and define, recursively, an+1 = n lnn (n + 1) Inn +2 -An, n ≥ 2. Prove that L = 1 in Raabe's test but Σ an converges by Kummer's test. Hint: let pn = n lnn, n ≥ 2 and use lim 1 nxo - n 1 n = e−1 Theorem 7.8 (Root test) Given the series Σ a,, let L = lim /\a,]. Then 8 1. a, converges absolutely if I < 1; n=1 8 2. a diverges if L > 1; n=1 3. the test is inconclusive if L = 1. n=1 841 n =] Theorem 7.9 (Raabe's test) If Σ a, is a series of nonzero real numbers with L= lim n(1a1/an), then 8 848 1. Σan converges absolutely if L > 1; n=1 8 2. Σa, diverges or converges conditionally if L < 1; n=1 3. the test is inconclusive if L = 1.