Let Hn Show that for n 1 H2n 1 n2SolutionHn summation k1

Let Hn = Show that for n 1, H2n 1 + n/2.

Solution

H_n = summation k=1 to n , 1/n

so H_2^n = 1/2^n

     summation n>=1 ,   H_2^n = 1/2 + 1/2^2 + 1/2^3......

r = 1/2

|Sn| =a/(1-r) = 1/2/[1 - 1/2] = 1/2/1/2 = 1

now H_{2^n =} 1 + [1/2 + 2/2 + 3/2 + 4/2...........n/2]

hence H_{2^n =} 1 + n/2

or H_2^n   >= 1 + n/2