Stuck on Applying Cauchy Convergence Criterion - Limit Theory I get stuck on the following problem for a rather long time that I finally decide to ask for help. The problem is as below:
Determine whether the sequence converges by applying Cauchy Convergence Criterion: $x_n = a_1*q+a_2*q^2+...+a_n*q^n$, where $|q|<1$ and $|a_i|<M$ for $i = 1,2,3,..$
For this question I have first thought of directly applying the Cauchy Convergence Criterion but with no luck . All I could get was only $x_n-x_m \leq |x_n-x_m| = M|q^n|+M|q^{n-1}|+...+M|q^{m+1}|$ for all $n,m \geq N$ (some $N \in \mathbb{N}$). This did not get me any further.
Then I considered the Geometric Sum Formula. I tried to use the formula to represent the sequence in its maximal form, i.e.: $q*M*(\frac{1-|q^n|}{1-q})$. But this was no good as well as when I wanted to establish the inequality, I realised $x_n-x_m$ is not necessarily smaller than the subtraction of my "maximal" sequences. I realize something goes wrong in my attempt, but I have yet figured out what that is. Can somebody please teach me how to deal with this problem? Thank you!

From where you left off: Assume $$n \geq m \Rightarrow  |x_n-x_m| \leq M\left(|q|^{m+1}+|q|^{m+2} + ...+ |q|^n\right)= M|q|^{m+1}\left(1+ |q|+ |q|^2+...+|q|^{n-m-1}\right)= \dfrac{M|q|^{m+1}\left(1-|q|^{n-m}\right)}{1-|q|} < \dfrac{M|q|^{m+1}}{1-|q|}$$. Thus let $\epsilon > 0$ be given, and choose  N$ \in \mathbb{N}$ such that if $m > N$ we have: $|q|^m < \dfrac{\epsilon\left(1-|q|\right)}{M|q|}$. Thus if $m,n > N \Rightarrow |x_n-x_m| < \epsilon$, and this shows $\{x_n\}$ is a Cauchy sequence.
