Question

2. Find upper and lower bounds for the following finite sums: (a) 1+1/V2+ 23) + 1/(1 + 33) + . . . + 1/(1 + n*). Try some variations of your own. 3. Use an e-N argument to find and prove the limn-o V5 + n2. Try some ations of your own.

Answer 1

(a) 1 + 1/squareroot 2 + 1/squareroot 3 + + 1/squareroot n 1 + 1 + + 1 greaterthanorequalto 1 + 1/squareroot 2 + 1/squareroot 3 + 1/squareroot n greaterthanorequalto 1/squareroot n + 1/squareroot n + + 1/1/squareroot n n greaterthanorequalto 1 + 1/squareroot 2 + 1/squareroot 3 + + 1/squareroot n greaterthanorequalto n/squareroot n = squareroot n (b) + 1 + 1 + 1 + + 1 greaterthanorequalto 1/1 + 1/2^3 + + 1/n^3 greaterthanorequalto 1/n^3 + 1/n^3 + + 1/n^3 n greaterthanorequalto 1/1 + 1/2^3 + + 1/n^3 greaterthanorequalto n/n^3 = 1/n^2