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Let \(\displaystyle I_1 = \int _0^1 \frac {e^x}{1+ x } dx\) and ...

\[\large \dfrac{1}{2} + \dfrac{1}{6} + \dfrac{1}{12} + \dfrac{1}{20} + \cdots =\ ?\]

\[\large \frac{-132!+131!-128!}{128!} \equiv x \text{ (mod 137)}\]

Find \(x\), where \(0 \le x \le 136\).

\[ 1+\dfrac{1}{3}+\dfrac{1}{3}\cdot\dfrac{3}{6}+\dfrac{1}{3}\cdot\dfrac{3}{6}\cdot\dfrac{5}{9}+\dfrac{1}{3}\cdot\dfrac{3}{6}\cdot\dfrac{5}{9}\cdot\dfrac{7}{12}+\cdots=2\cos\theta\]

Let \(y = \sin^2(x)\). Find \(\dfrac{d^2y}{dx^2}\).

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