Let α ∈ 0 , 1 and β = log e 1 - α . Let P n x = x + x 2 2 + x 3 3 + … . + x n n , x ∈ 0 , 1 . Then the integral ∫ 0 α t 50 1 - t d t is equal to

Question

Let &#945; &#8712; 0 , 1 and &#946; = log e 1 - &#945; . Let P n x = x + x 2 2 + x 3 3 + &#8230; . + x n n , x &#8712; 0 , 1 . Then the integral &#8747; 0 &#945; t 50 1 - t d t is equal to

Aicharya · Accepted Answer

Let I = &#8747; 0 &#945; t 50 1 - t d t &#8658; I = - &#8747; 0 &#945; 1 - t 50 - 1 1 - t d t &#8658; I = - &#8747; 0 &#945; 1 - t 50 1 - t d t + &#8747; 0 &#945; 1 1 - t d t &#8658; I = - &#8747; 0 &#945; 1 + t + t 2 + . . . + t 49 d t - ln 1 - t 0 &#945; &#8658; I = - t + t 2 2 + t 3 3 + . . . + t 50 50 0 &#945; - ln 1 - &#945; &#8658; I = - &#945; + &#945; 2 2 + &#945; 3 3 + . . . + &#945; 50 50 - ln 1 - &#945; &#8658; I = - &#946; + P 50 &#945;

Let α ∈ 0 , 1 and β = log e 1 - α . Let P n x = x + x 2 2 + x 3 3 + … . + x n n ,…

Solution

Let &#945; &#8712; 0 , 1 and &#946; = log e 1 - &#945; . Let P n x = x + x 2 2 + x 3 3 + &#8230; . + x n n ,…

Solution

Let α ∈ 0 , 1 and β = log e 1 - α . Let P n x = x + x 2 2 + x 3 3 + … . + x n n ,…