Prove that `.^(n)C_(1) - (1+1/2) .^(n)C_(2) + (1+1/2+1/3) .^(n)C_(3) + "…" ``+ (-1)^(n-1) (1+1/2+1/3 + "…." + 1/n) .^(n)C_(n) = 1/n`

Explanation

To prove the equation:

\[ \binom{n}{1} - \left(1 + \frac{1}{2}\right) \binom{n}{2} + \left(1 + \frac{1}{2} + \frac{1}{3}\right) \binom{n}{3} + \ldots + (-1)^{n-1} \left(1 + \frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{n}\right) \binom{n}{n} = \frac{1}{n} \]

we will follow these steps:

...