How to show that $$\sum_{n=1}^{\infty}\alpha_nq^n$$is convergent using Cauchy condition for uniform convergence of series? $|q|<1,|\alpha_n| $$\forall\varepsilon>0,\exists\overline n,\forall n>\overline n,\forall k\in \Bbb{N} \left |\sum_{i=n}^{n+k}a_i\right|<\epsilon$$ I think it should have been done by taking $\varepsilon>0$ and then $\overline n>(\text{something dependent from }\varepsilon ) $ But the one idea I had was to take geometric series and it is convergence but this won't be from that definition I guess.
show that $\sum_{n=1}^{\infty}\alpha_nq^n$ is convergent using Cauchy condition for uniform convergence of series
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sequences-and-series
convergence
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$$ \Bigl|\sum_{i=n}^{n+k}\alpha_i\,q^i\Bigr|\le M\sum_{i=n}^{n+k}|q|^i\le M\sum_{i=n}^{\infty}|q|^i=M\frac{|q|^n}{1-|q|}. $$