p-1 mod 4, prove that Σ k ( )-0. Let p be an odd prıme. Suppose that p k=1 p-1 mod 4, prove that Σ k ( )-0. Let p be an odd prıme. Suppose that p k=1
Find the Maclaurin series of 2a. Σ Preview η =0
Exercise 1 (pts 5). Prove that Σσι(α) = ” + Οζω log(n). . - Τ. η<α π2 We recall that Σ
Prove that (n + m r) = Xr k=0 (n k) (m r − k) . (Here r ≤ n and r ≤ m.) Probability theory by Dr Nikolai Chernov
(a) Starting with the geometric series X?, find the sum of the series η ΕΟ Σ ηχο – 1, 1x] <1. ΠΕ 1 (b) Find the sum of each of the following series. DO Σηχή, 1x <1 η = 1 η (i) Σ. (c) Find the sum of each of the following series. D) Σπίη – 1)x, Ix <1 ΠΕ 2 (i) Σ - η 57 ΠΕ 2 0 i) 22 = 1
η = 36 H 0 1 220 και = 18 Haμ « 20 σ = 12 The test statistic equals Ο 1.30 -1.00 -1.30 Ο 1.50
find the radius of convergence 2) Σ(*) k-kak 11 k=0. k=0 24 b) Σ d) Σ (1+ (*). Υ k=1 k=1
a2 (a) Prove that g converge uniformly to 0 on (O, M for any M>0, but does pot converge uniformly to 0 on (0, oo) (b) Prove that 19 converges uniformly on [Q M for any M>0 Does Σ"-1 g" converge uniformly on (, x)? Does Σ"-1 g" define a continuous function on (, x)? ii. iii.
Σ(6-3-1) η-1
2. Test the Series for convergence or divergence. In(n) Σ(-) Σ- 4 n=3 η=1 n 3. Determine which option is absolutely converges and explain in details the reason. 1 (=Σ(-1)" 3 =Σ(-1)" C-Σ(-1)* tan(n) η Υ -Σ-1): E = None of these n!