Determine whether the statement is true or false.
Let $c\in C,$ and $n\in \Bbb N$ \ {$0$}. Suppose $|c|\gt5$. Then $|\sum_{j=0}^n \frac{5^j}{c^j}-\frac{c}{c-5}|\leq\frac{(5^{n+1})}{|c|^n(|c|-5)}$.
As the LHS looks like a geometric series, so I tried to get a common ratio $r=1+\frac{5(c-5)^2}{c(c^2-5c+25)}$.(I'm not sure did I calculate correctly tho) However it seems that it's not useful. Or it actually does not related to sum of geometric sequence? Any help/hints to start the proof? Any inequalities can be applied? Thanks.
$\sum_{j=0}^n \frac{5^j}{c^j}=\frac {1-(5/c)^{n+1}} {1-5/c}$ by the formula for geomertic sum with common ratio $5/c$. So the left side of the inequality is $|\frac {1-(5/c)^{n+1}} {1-5/c}-\frac c {c-5}|$ which simplifies to $|\frac {c^{n+1}-5^{n+1}}{c^{n}(c-5)}-\frac {c^{n+1}}{c^{n}(c-5)}|=|\frac {5^{n+1}}{c^{n}(c-5)}|$. I will let you finish using the fact that $|c-5| \geq |c|-5$.