Find the sum of the following.

up to n terms.

The given sum can be written as (1 + 1 + 1 + …) - (1/n, 2/n, 3/n, …)

Sum of first series up to n terms = 1 + 1 + 1 + … up to n terms

= n

Now, consider the second series:

Here, first term = a = 1/n

Common difference = d = (2/n) - (1/n) = (1/n)

Now, Sum of n terms of an arithmetic series is given by:

S_n = [2a + (n - 1)d]

Therefore sum of n terms of second arithmetic series is given by:

∴ S_n = [2(1/n) + (n - 1)(1/n)]

= [(2/n) + 1 - (1/n)]

= [(1/n) + 1]

= = = (n + 1)/2

Now, sum of n terms of the complete series = Sum of n terms of first series - Sum of n terms of second series

= n - (n + 1)/2

= (2n - n - 1)/2

= 1/2 (n - 1)