Convergency and Divergency!

Calculus Level 5

$\large I(n) =\int_0^1 \dfrac{\ln(1-t)}{(1-t)^n} \, dt \qquad \qquad G(n) = \int_0^1 \dfrac{\ln(1+t)}{(1+t)^n} \, dt$

Let $$I(n)$$ and $$G(n)$$ be functions as shown above. Which of the following statements is/are true?

(A): $$G(n)$$ is convergent for all $$n$$.

(B): I(n) is divergent for n>1

(C): $$I(n) + G(n)$$ is convergent for all $$n$$ < 1.

(D): $$\displaystyle e^2 = \left( \left( \sum_{r=0}^\infty \frac1{I^{(r)}(2)} \right)+ 1 \right)^{-1}$$, where $$I^{(r)} (\cdot)$$ denotes the $$r^\text{th}$$ derivative of $$I$$.

(E): $$I(n) : [1,\infty) \to \mathbb R$$ is a bijective function.

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