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## Induced morphism between image and ker (category theory)

Context question: “Differential on a object of an abelian category” Let $C$ be an abelian category and $x$ an object of $C$. Consider a morphism $f: x\to x$ such that $f\circ f=0.$ I can see two ways to define a morphism $\text{im}\, f\to \ker f:$ Consider the diagram where the morphisms $i$ and $j$ are […]

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can someone explain to me why addition is countable when it is a finite sequence of terms separated by + signs: i.e.(x1 + x2 + ….+xk) or (13, 2 + 3 + 3, 10 + 11 + 12) because i know to be countable it needs to have a one-to-one correspondence with Natural numbers. But […]

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## Is there a series representation for $\frac{1}{\log(x)}$?

for $x>0$ it is known that: $$\log(x)=2\sum_{k=1}^\infty \frac{\frac{x-1}{x+1}^{2k-1}}{2k-1}$$ Is there a series representation for $\frac{1}{\log(x)}$ in the following form? $$\frac{1}{\log(x)}=a_1(x)+a_2(x)+a_3(x)+\dots=\sum_{k=1}^\infty a_k(x)$$ Or, is there a way to transform the expression $$\frac{1}{2\sum_{k=1}^\infty \frac{\frac{x-1}{x+1}^{2k-1}}{2k-1}}$$ into the aforementioned form?

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