Show that: C₀ - C₁ (1+x / 1+nx) + C₂ [1+2x / (1+nx) ²] - C₃ [1+3x / (1+nx) ³] + ……..= 0

Find the sum of the series: ∑ (-1) ^r. ⁿC_r (1/2^r + 3^r/2^2r + 7^r/2^3r + …… + up to m terms) r goes from 0 to n.

Dear Prathamesh,

In order to get the sum of the series you have to provide the first three terms , then only one can understand what is the nature of the series. For e.g, if I give a series as follows,

2 , 4 , .......................................................... till n^th term

Now can you tell me is it in G.P or A.P or A.G.P or H.P

Ofcourse not! So this is the problem in the question:

8/5 + 16/65 + ..................................+ 128/2¹⁸ + 1 .