By considering (1 + x)^(n + 1), show that 1 + (1/2)*n!/[1![(n - 1)!] + (1/3)*n!/[2![n - 2)!] + ... + 1/(n + 1)?

For positive integer values of 'n'
... ..... .... n
(1 + x)ⁿ = Σ ⁿCr x^r
... .... ... r = 0

From now on I am not going to write the limits for the summation.

(1 + x)ⁿ = Σ ⁿCr x^r

Integrating both sides w.r.t x

∫ (1 + x)ⁿ dx = ∫ Σ ⁿCr x^r dx

{(1 + x)^(n+1)}/(n+1) = Σ ⁿCr (∫ x^r dx) + k [k = arbitrary integration constant]

{(1 + x)^(n+1)}/(n+1) = Σ ⁿCr [{x^(r+1)}/(r+1)] + k

If the upper and lower limits are applied to the integration as from x = 0 to x = 1

{(1 + 1)^(n+1)}/(n+1) - {(1 + 0)^(n+1)}/(n+1) = Σ ⁿCr [{1^(r+1)}/(r+1)] - Σ ⁿCr [{0^(r+1)}/(r+1)]

{2^(n+1)}/(n+1) - 1/(n+1) = Σ ⁿCr {1/(r+1)} - 0

{2^(n+1) - 1}/(n+1) = Σ ⁿCr {1/(r+1)}

Hence proved.

Hope this helps!