Let $m,k\in\mathbb{N}^*,\displaystyle P_k=\prod_{u=1}^m\dfrac{\text{Li}_{2u+1}(k)}{\text{Li}_{2u}(k)}$
Is there a way to simplify this product ?
What is its behavior for $k\rightarrow\infty$, for $m\rightarrow\infty$, and for $k,m\rightarrow\infty$ ?
This would help me to solve a sum involvi...
In mathematics, the rank–nullity theorem of linear algebra, in its simplest form, states that the rank and the nullity of a matrix add up to the number of columns of the matrix. Specifically, if A is an m-by-n matrix (with m rows and n columns) over some field, then
This applies to linear maps as well. Let V and W be vector spaces over some field and let T : V → W be a linear map. Then the rank of T is the dimension of the image of T and the nullity of T is the dimension of the kernel of T, so we have
or, equivalently,
One can refine this statement (via the splitting lemma or the below proof...
