# Do the zeroes of this polynomial lie inside, outside, or on the unit circle? $P_n(z)=1^3 z + 2^3 z^2 + 3^3 z^3 + \cdots + n^3 z^n$

For each positive integer $$n$$, let's define the polynomial $$P_n(z)=1^3 z + 2^3 z^2 + 3^3 z^3 + \cdots + n^3 z^n$$ Do the zeroes of $$P_n$$ lie inside, outside, or on the unit circle $$|z|=1$$?

I tried to find a formula for $$\displaystyle \sum_{k=1}^n k^3 z^k$$ by repeatedly taking derivatives of $$z^k$$ but it was so tough. Initial investigation showed that the zeroes lie inside the unit circle but I couldn't generalize the result.

Any help would be appreciated!

Source : The Arts and Crafts of Problem Solving

Let $$g(z) = \sum_{k=0}^n z^k = \frac{z^{n+1}-1}{z-1}$$

It is easy to see the roots of $$g(z)$$ lie on the unit circle $$|z|= 1$$ and all of them are simple.

By Gauss-Lucas theorem, the roots of $$g'(z)$$ belong to the convex hull of the roots of $$g$$. Since the closed unit disk $$|z| \le 1$$ is convex, this convex hull is a subset of the closed unit disk. Notice the convex hull is a $$n$$-gon which intersect the unit circle $$|z| = 1$$ only at the roots of $$g(z)$$. Since the roots of $$g(z)$$ are simple, none of them can be root of $$g'(z)$$. As a result, the roots of $$g'(z)$$ belongs to the open unit disk $$|z| < 1$$.

Since $$zg'(z)$$ differs from $$g'(z)$$ by only a root at $$0$$, the roots of

$$zg'(z) = \sum_{k=1}^n k z^k$$

belong to the open unit disk $$|z| < 1$$.

Apply Gauss-Lucas theorem again and then add a root at $$z = 0$$, we find the roots of

$$\left(z\frac{d}{dz}\right)^2 g(z) = \sum_{k=1}^n k^2 z^k$$ belong to the open unit disk $$|z| < 1$$. Repeat this process one more time, we find all the zeros of

$$\left(z\frac{d}{dz}\right)^3 g(z) = \sum_{k=1}^n k^3 z^k = P_n(z)$$ lie inside the unit circle.

Notes

There are other ways to arrive at same conclusion. In particular, we can use following results:

Let $$\displaystyle\;f(z) = \sum_{k=0}^m a_k z^k$$ be any polynomial with real and positive coefficients.

1. If the coefficients $$a_k$$ are non-descending, $$0 < a_0 \le a_2 \le \cdots \le a_m$$ then roots of $$f(z)$$ belong to the closed unit disk $$|z| \le 1$$.

2. If the coefficients $$a_k$$ are non-ascending, $$a_0 \ge a_1 \ge \cdots \ge a_m > 0$$ then roots of $$f(z)$$ lie outside the open unit disk (i.e. $$|z| \ge 1$$ for all the roots)

3. In general, the roots of $$f(z)$$ belong to the closed annulus $$\min_{1\le k \le m} \frac{a_{k-1}}{a_k} \le |z| \le \max_{1 \le k \le m}\frac{a_{k-1}}{a_k}$$

Since it is easy to derive any one of these results from the other two, these results are typically treated as a single theorem known as the Eneström-Kakeya Theorem.

For a proof of the first result, see answers of a related question. In particular, the answer by Ayman Hourieh which uses Rouché's theorem.

Back to the problem at hand. It is easy to see we can rewrite $$P_n(z)$$ as $$z f(z)$$ for some polynomial $$f(z)$$ with real and positive coefficients. Apply the third result, we immediately find aside from a root at $$z = 0$$, the remaining $$n-1$$ roots of $$P_n(z)$$ lies within the closed annulus $$\frac18 \le |z| \le \left(\frac{n-1 }{n}\right)^3 < 1$$