Let α ∈ (0, 1) and β = _ e (1 - α) . Let P _ n (x) = x + x ^ 2 2 +...

Let

$I = {∫}_{0}^{\alpha}\dfrac{{t}^{50}}{1 - t}dt$

$⇒ I = - {∫}_{0}^{\alpha}\left(\dfrac{1 - {t}^{50} - 1}{1 - t}\right)dt$

$⇒ I = - {∫}_{0}^{\alpha}\left(\dfrac{1 - {t}^{50}}{1 - t}\right)dt + {∫}_{0}^{\alpha}\left(\dfrac{1}{1 - t}\right)dt$

$⇒ I = - {∫}_{0}^{\alpha}\left(1 + t + {t}^{2} + . . . + {t}^{49}\right)dt - {\left[\ln \left(1 - t\right)\right]}_{0}^{\alpha}$

$⇒ I = - {\left[t + \dfrac{{t}^{2}}{2} + \dfrac{{t}^{3}}{3} + . . . + \dfrac{{t}^{50}}{50}\right]}_{0}^{\alpha} - \ln \left(1 - \alpha\right)$

$⇒ I = - \left[\alpha + \dfrac{{\alpha}^{2}}{2} + \dfrac{{\alpha}^{3}}{3} + . . . + \dfrac{{\alpha}^{50}}{50}\right] - \ln \left(1 - \alpha\right)$

$⇒ I = - \left(\beta + {P}_{50}\left(\alpha\right)\right)$