Let $\mathbf{x} \in \mathbb{R}^{n} \setminus \{\mathbf{0}\}$. Let $m \in \mathbb{N}$. I'm looking at the following summation:
$\sum\limits_{j=1}^{n} (-1)^{j-1} \frac{ || \mathbf{x} ||^{2} - m x_{j}^{2} }{||\mathbf{x}||^{m+2}}$
I'm wondering, is there a way to write the above summation more concisely?
In addition to this, for what values of $m$ does the above sum to $0$?
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So far, this is what I have computed:
$\sum\limits_{j=1}^{n} (-1)^{j-1} \frac{ || \mathbf{x} ||^{2} - m x_{j}^{2} }{||\mathbf{x}||^{m+2}} = || \mathbf{x} ||^{-m-2} \sum\limits_{j=1}^{n} (-1)^{j-1}\left[ || \mathbf{x} ||^{2} - m x_{j}^{2} \right] \\ = || \mathbf{x} ||^{-m-2} \sum\limits_{j=1}^{n} (-1)^{j-1}\left[ \left( \sum\limits_{\ell=1}^{n} x_{\ell}^{2} \right) - m x_{j}^{2} \right] \\ = || \mathbf{x} ||^{-m-2} \sum\limits_{j=1}^{n} \sum\limits_{\ell=1}^{n} (-1)^{j-1}\left[ x_{\ell}^{2} - \frac{m}{n} x_{j}^{2} \right]$
I am stuck from here.
(For anyone interested, this is a portion of Problem 30.6 in Munkres' Analysis on Manifolds)