Question

Among the following series, which ones "converge conditionally"? Please need a complete answer. Thanks!

Σ(-1)-n(n+2) Σ(-1)"sini COS Σ(-1)"n'tanî n2 Σ(-1)nne'n 1,13 1-3.5 3 3.5 3.57++C-1)" 1-3.5.(2n-1 3.5.7 (2n 1) C-1 2n) 1.3.5 (2n-1)

Answer 1

sigma_n = 1^infinity (- 1)^n 1/squareroot n (n + 2) = sigma_n = 1^infinity (- 1)^n u_n which is an alternating series (A, S) Here u_n = 1/squareroot n (n + 2) we choose v_n = 1/n Now, lt_n rightarrow infinity u_n/v_n = lt_n rightarrow infinity 1/squareroot n (n + 2)/1/n = lt_n rightarrow infinity n/n squareroot 1 + 2/n = 1 (Notequalto 0 and finite) By p-series Test, sigma v_n = sigma 1/n is divergent and By comparison Test, sigma u_n and sigma v_n we have a like Therefore sigma_n = 1^infinity (- 1)^n 1/squareroot n (n + 2) is conditionally convergent (2) sigma_n = 1^infinity (- 1)^n ln (n + 1/n) = sigma_n = 1^infinity (- 1)^n u_n which is an A S Here u_n = l_n (n + 1/n) greaterthanorequalto 1/2n Therefore By p-series Test, sigma u_n = sigma u_n = sigma 1/2n is divergent Therefore sigma_n = 1^infinity (- 1)^n ln (1 + n/n) is conditionally convergent