Please answer all questions and show work so I can understand

1. Use sigma notation to rewrite the following infinite series. Starting with n = 1. 1 (a) 1- + 3 4 + ... (b) 1-1+1-1+1-1+ . .. 1 2 3 4 (c) it - 4 + 8 16 + .. 2 2. Compute the first 4 partial sums $1, $2, $3, S, for the following series. (a) n=1 (b) sin 2 n=1 3. State whether the given geometric series converges or diverges and explain why. If it is convergent, please find its sum S. 1 (a) 1+ 10 100 1000 2 e (b) 1+-+ 7 .. -2 (c) 1- 091 81 -- 271. Use the Divergence Test to test the divergence for the following series. If the divergence test doesn't apply, state why. (a) n + 2 n= 1 (b) 2n3 + 1 (2n + 1)(n - 1) (c) (n + 1) (d) e - 2 2. Use either the (Direct) Comparison Test or the Limit Comparison Test to determine the convergence of the series. (a) n+1 2n3 -1 n+1 [b) > n=1 ny 8 n- + 5n (c) 8 (d) > 102 n= 1 (e) an + 4n n + 6n 3. For the following series E n=1 n + 2 (a) Use the Integral test to test the convergence (b) Use the Limit Comparison Test to test the convergence