Question

(a) State the First Comparison Test and show that the following series con- verges: O0 1 + cos ((2n +1)!) (b) Determine whether the following series converges (c) State the Integral Test and sketch its proof (d) Prove or disprove: If a series Σ001 an converges then Σηι an converges absolutely. e) Answer the following two questions without proof: For which r E R is the geometric series 0O convergent? What is the limit of the series in case of convergence?

Answer 1

a) sigma^infinity_n = 2 squareroot 1 + cos (2n + 1)! /n^4 + squareroot 1 + 5m^6 we know cos (2n + 1)! lessthanorequalto 1 Forall n elemenetof N So squareroot 1 + cos (2n + 1)!/n^4 + squareroot 1 + 5n^6 lessthanorequalto 2/n^4 + squareroot 1 + 5n^6... (1) Let a_n = 2/m^4 + squareroot 1 + 5n^6 b_n = 1/n^4 lim_n rightarrow infinity a_n/b_n = 2n^4/n^4 + squareroot 1 + 5n^6 = 2 Notequalto 0 So by limit comparison test sigma a_n & sigma b_n converges or diverges together. But sigma b_n is c g t (by P - test) implies sigma 2/squareroot n^4 + 5n^6 is convergent implies use this in (1) we get our series is convergent by comparison test. So, series is convergent \r\n sigma^infinity_n = 1 1/squareroot n^2 + 1 - squareroot n^2 - 1 = sigma^infinity_n = 1 squareroot n^2 + 1 + squareroot^n^2 - 1/2 (Rationalize) let b_n = n a_n = squareroot n^2 + 1 + squareroot n^2 -1/2 .............