So I am asked to find the Taylor polynomial of $f(x) = \frac{1}{x}$ about the point $a=1$ for ever n$\in{N}$, and then use induction to justify the answer. 

I got the Taylor polynomial which was simple enough:

$$T\_{n}(x)=\sum\_{n=0}^{\infty} \frac{f^n(x)(a)}{n!}(x-a)^n$$

$$f(x) = 1-(x-1)+(x-1)^2-(x-1)^3+(x-1)^4+...$$

$$T\_{n}(x)=\sum\_{n=0}^{\infty} (-1)^n(x-1)^n$$

That wasn't too bad. How do I justify this with induction though? I am a little confused as to how I would start this.

I tried writing out the terms as such:

$$1-(x-1)+(x-1)^2-(x-1)^3+(x-1)^4+...+(-1)^k(x-1)^k = \frac{1}{x}$$

I am not really sure how to go about doing this though... Is my step valid? I would appreciate is someone could guide me in the right direction.