Determine whether the infinite series is convergent or divergent. If the series converges, find its sum. 1. X[infinity] n=1 1 + sin3 n n3 + 5 2. X[infinity] n=1 n + 3n 2 √ 1 − n2 + n5 3. X[infinity] n=1 (−3)n + 2n 4 n 4. X[infinity] n=0 p 4n2 + 1 − n

Type the correct answer in each box. use numerals instead of words. if necessary, use / for the fraction bar. consider the given table.

When 142 is added to a number the result is 64 more then 3 times the number. option 35 37 39 41

Prove (a) cosh2(x) − sinh2(x) = 1 and (b) 1 − tanh 2(x) = sech 2(x). solution (a) cosh2(x) − sinh2(x) = ex + e−x 2 2 − 2 = e2x + 2 + e−2x 4 − = 4 = . (b) we start with the identity proved in part (a): cosh2(x) − sinh2(x) = 1. if we divide both sides by cosh2(x), we get 1 − sinh2(x) cosh2(x) = 1 or 1 − tanh 2(x) = .