Proving that this pair of series' can only be satisfied with infinitely many terms.

Question

I'm trying to prove that the following pair of series can only be satisfied if we have infinitely many coefficients.

$$k^2 = \sum_{n=1}^{M} a_n k^n \text{ ,} $$ $$ k^3 = \sum_{n=1}^{M} (-1)^na_n k^n $$

where $k$ is a positive integer.

Answer 1

You might want to also exclude the $k=1$ case.

Let $k=1$ and let

$$a_n=\begin{cases} 0&\text{ for }n\text{ odd}\\ \frac{1}{j-1}&\text{ for }n\text{ even and }M=2j-1\text{ odd}\\ \frac{1}{j}&\text{ for }n\text{ even and }M=2j\text{ even} \end{cases}$$

Then if $M=2j-1$, each $a_n=\frac{1}{j-1}$ and

$$ \sum_{n=1}^{2j-1} a_n k^n=\frac{1}{j-1}(1^2+1^4+\cdots+1^{2j-2})=1=1^2 $$

And also

$$\sum_{n=1}^{2j-1} (-1)^na_n k^n=\frac{1}{j-1}(1^2+1^4+\cdots+1^{2j-2})=1=1^3 $$

If $M=2j$ then

$$ \sum_{n=1}^{2j} a_n k^n=\frac{1}{j}(1^2+1^4+\cdots+1^{2j})=1=1^2 $$

and

$$\sum_{n=1}^{2j} (-1)^na_n k^n=\frac{1}{j}(1^2+1^4+\cdots+1^{2j})=1=1^3 $$