I can show that each of the matrices is positive (by which I mean positive semi-definite). You would then need to show that the series converges, and it would follow that is positive (because positivity is preserved by taking sums and limits).

Since the numerical coefficient (in the definition of ) is clearly positive, you are trying to show that the matrix is positive for a certain value of , where

.

(Here, I have used in place of your , and I have used the fact that to rewrite the entries below the diagonal.)

In fact, the matrix is positive for all values of . The quickest way to see this is to use complex numbers and to write . Then the complex conjugate and the inverse of are equal: . Also, . Define a matrix by . Then is an matrix whose -element is (for and ). The matrix has -element which is equal to the -element of .

Therefore . But any matrix of the form is positive, so it follows that is positive.