Prove that if n is a positive integer then Prove that if n i

Prove that, if n is a positive integer, then

Prove that, if n is a positive integer, then Summation_i = 1^n 1/i(I + 1) = 1 - 1/n + 1

Solution

sigma 1/i(i+1)

so, in the numerator add i and substract i

sigma 1+i-i/i(i+1)   = sigma 1+i/i(i+1) -      sigma i/i(i+1)

=sigma 1/i - sigma 1/(i+1)

1+1/2+1/3+1/4.... 1/n         - [1/2+1/3+1/4....... 1/n+1/(n+1)]

1-1/(n+1)

hence,proved