1. n2 < 2" forn 4,5, 6,... You may assume: 2n +1 < 2" for n = 4, 5, 6,... 2. 13+ 23 + 33+...+n = "(n+1) for all n = 1,2,3,... 4 3. ++... + > Vn for n = 2, 3, 4,... 4. II, = ntl for n = 2, 3, 4, ... (The symbol II means product) %3D 2n 5. E1 = for n = 1, 2, 3... %3D 412-1 2n+1 6. 1+3+5+ ..+ (2n 1) = n2 for all n = 1,2, 3,... 7. If r # 1, then 1+r+x2 + x3 + x + . .. + x" = l-z**) for all n 1-r(n+1) 1-r 30,1, 2, 3, ... 8. 1+5+9+ ..+ (4n 3) = 2n2- n for all n = 1, 2, 3,... 9. In the first two problems, D" (f (x)) stands for the n-th derivative of the function f at r. Also you may assume the product rule for derivatives, that D'r' = Dx = 1, and that r = 1. %3D (a) Prove : D" (r") = n! for n 1,2, 3,... = nx"- for n 1,2, 3,... %3D (b) Prove: D (1") %3D (c) Prove: (cos +i sin r)" = cos nx +i sin nx for n 0,1,2,... Here i? = -1. = (d) Assume that z> -1. Prove that (1+x)" >1+ nx forn=0, 1, 2,.... ......