Question

Binomial Expansion Series $ \left( \frac{(1 + x)}{(n + 1)} \right)' = n_0 x + n_1 \frac{x^2}{2} + n_2 \frac{x^3}{3} + \cdots + n_n \frac{x^n}{n+1} $

Hint:

The binomial series is an approximation for powers of \( (1 + x) \), and the general form of each term involves binomial coefficients along with increasing powers of \( x \).

Answer 1

This expression is an expansion of the binomial expression \( \left( \frac{(1 + x)}{(n + 1)} \right)' \), where the right-hand side represents the binomial expansion of the series for \( (1 + x)^{n+1} \).
Each coefficient \( n_k \) corresponds to the binomial coefficient in the expansion. As we expand \( \left( \frac{(1 + x)}{(n + 1)} \right) \), the result includes terms like:
- The first term \( n_0 x \) is simply \( x \).
- The second term is \( n_1 \frac{x^2}{2} \), which is \( \frac{x^2}{2} \), and so on.
The general term in this expansion is \( n_k \frac{x^k}{k+1} \), where \( n_k \) are binomial coefficients.
This expansion allows us to approximate the behavior of \( \left( 1 + x \right)^{n+1} \) for small values of \( x \).