Let $(b_n)_{n \geq 0}$ a sequence of complex numbers such that $\sum_{n=0}^{\infty} |b_n| < \infty$. Let's suppose also that $$ \sum_{n=0}^{\infty} \frac{b_n}{k^n}=0 $$ For each integer $k \geq 2$. I want to show that $b_n=0$ for all $n$.
My attempt:
If $(z_n)$ is any sequence of complex numbers we know that $z_n \rightarrow z$ if and only if $Re (z_n) \rightarrow Re (z)$ and $Im (z_n) \rightarrow Im (z)$. Therefore, it's enough to show the result in the case when $b_n \in \mathbb{R}$. Let's assume then that $b_n \in \mathbb{R}$. Since $\sum_{n=0}^{\infty} |b_n| < \infty$ it follows that the series formed from its positive terms and the series formed from its negative terms both converge. We can then write $b_n=b_n^+ - b_n^{-}$ for all $n$, and $(b_n^+), (b_n^{-})$ are two sequences of non-negative numbers such that their respective series converge.
Since $b_n=b_n^+ - b_n^{-}$, we have then that: $$ \left| \sum_{j=0}^{n} \frac{b_j}{k^n} \right| \geq \left| \left| \sum_{j=0}^{n} \frac{b_j^{+}}{k^n} \right| -\left| \sum_{j=0}^{n} \frac{b_j^{-}}{k^n} \right| \right| \geq0 $$ Therefore, since the sequence of the left converges to $0$ and $\sum_{j=0}^{n} \frac{b_j^{+}}{k^n} ,\sum_{j=0}^{n} \frac{b_j^{-}}{k^n} $ both converge, we have that $$ \left| \left| \sum_{j=0}^{n} \frac{b_j^{+}}{k^n} \right| -\left| \sum_{j=0}^{n} \frac{b_j^{-}}{k^n} \right| \right| \rightarrow 0 \textrm{ as } n \rightarrow \infty $$
Which implies that $$ \sum_{j=0}^{\infty} \frac{b_j^{+}}{k^n} =\sum_{j=0}^{\infty} \frac{b_j^{-}}{k^n} $$ Of course, from here I'd love to conclude that there exist $N \in \mathbb{N}$ such that $b_j^{+}=b_j^{-}$ for $n \geq N$ which would basically finish the proof. However, this doesn't follow necessarily from the last iquality since we can have two sequences of non-negative numbers with the same sum but which are not eventually equal as I suggested above. In other words, in general $$ \sum_{j=0}^{\infty}a_j =\sum_{j=0}^{\infty} b_j $$ doesn't imply that the sequences are almost equal. As you can see I am missing something.
How can I conclude/fix this issue?
In advance thank you very much.
Edit: I am looking to solve this exercise using the basics about sequences and series (or even known tools of real analysis). There is probably an easier solution using more advanced tools of complex analysis but I haven't seen them yet.