I consider the following error term where $\lambda\in(0,1)$. I want to find the value of $\lambda$ for which it is negative
$$ \lambda^{n-1} -2\sum_{k=n}^{\infty}\lambda^k $$
We have that
$$ \lambda^{n-1} -2\sum_{k=n}^{\infty}\lambda^k = \lambda^{n-1} -2\lambda^{n}\sum_{k=0}^{\infty}\lambda^k = \lambda^{n-1} - 2\frac{\lambda^n}{1-\lambda} $$
This to be negative implies
$$ \lambda^{n-1} < 2\frac{\lambda^n}{1-\lambda}\implies 1-\lambda < 2\lambda\implies \frac{1}{3} < \lambda $$
I think that my computation is correct, but I should find that it works for $\lambda$ close enough to $1$, which is not the case here ?
NB : close enough is not very precise but it is the word use in the exercise..
Thank you a lot for your help