Check whether this sequence is Cauchy or not

Question

Let $|q|<1$. Suppose that there is $M<∞$, such that for all $k∈N$, $|a_k| ≤ M$, $x_n =a_0 +a_1q+a_2q^2 +\cdots+a_nq^n$.

I know for a sequence to be Cauchy $|x_n-x_m| < \epsilon$ for all $m,n>N$.

This question had 2 other parts that I already proved, but I don't understand this question. I don't know how to start Anything to help me start would be great.

Answer 1

For some $m,n>N$ where $n>m$.

$$\begin{align} &|x_n-x_m| \\ &=(a_0 +a_1q+a_2q^2 +\cdots+a_nq^n)-(a_0 +a_1q+a_2q^2 +\cdots+a_mq^m) \\ & =(a_{m+1}q^{m+1} +a_{m+2}q^{m+2}+a_{m+3}q^{m+3} +\cdots+a_nq^n) \\ & \le (Mq^{m+1} +Mq^{m+2}+Mq^{m+3} +\cdots+Mq^n) \\ & =M(q^{m+1} + q^{m+2}+ q^{m+3} +\cdots+ q^n) \\ & =M\cdot q^{m+1}\cdot \left(\frac{1-q^{n-m}}{1-q}\right)\end{align}$$

Can you find the upper bound now?

Hope this helps.