Find the sum of the following series upto n terms

     =

     =

Let = ……(i)

⇒ 1 = A(n + 1) + Bn ……(ii)

[On multiplying both sides by n(n+1)]

To find A : Putting n = 0 in (ii), we get,

1 = A(0 + 1) ⇒ A = 1

To find B : Putting n = 0 in (ii), we get

1 = B(-1) ⇒ B = -1

Putting these values of A and B in (i), the partial fractions are

=

or T_n =

Putting n = 1, 2, 3,……n, we get,

T₁ =

T₂ =

T₃ = ………………….

T_n =

Adding vertically, we get,

S_n = 1 - =

Extra. Hence find sum to infinity:

Now, S_n = =

As n → ∞

or S∞ = = 0