Calculation of coefficient of x

Question

What is the coefficient of $x$ in $x^{-1}$ ? This question has been asked in an interview.

Answer 1

HINT:

Provided $|1-x| \le 1$ i.e. $0 < x \le 2$, we can say $$x^{-1}$$ $$=\frac{1}{x}$$ $$=\frac{1}{1-(1-x)}$$ $$=1+(1-x)+(1-x)^2+(1-x)^3+ \ldots$$

Now compute the coefficient of $x$ from the above expansion.

The answer will be

$$-\lim_\limits{n\to \infty}\sum_{r=1}^{n} r$$