Generalized Binomial Expansion

Discrete Mathematics Level 3

If the function \(f(x) = \sqrt{1 + x}\) is expanded out in terms of powers of \(x\) such that \(f(x) = \sum\limits_{k=0}^{\infty} a_{k} x^{k},\) what's the coefficient \(a_{3}\) of the \(x^{3}\) term?

Express your answer as an exact decimal.