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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