For each positive integer n , define f _ n ( x )= minimum ( x ^ n n ! , (1- x )^ n n ! ) , for 0 x 1 . Let I _ n = ₀¹ f _ n ( x ) dx , n 1 . Then I_ n = _ n=1 ^ I_ n is equal to-

Question

Quantrex Academy · Accepted Answer

Correct answer: A. 2 e -3

I_ n = ₀^ 1 / 2 x^ n n ! d x+ _ 1 / 2 ¹ (1-x)^ n n ! d x= 1 (n+1) ! ( ( 1 2 )^ n+1 + ( 1 2 )^ n+1 )= ( 1 2 )^ n (n+1) ! _ n=1 ^ I_ n = ( 1 / 2 2 ! + (1 / 2)² 3 ! + )=2 e -3

For each positive integer n , define f _ n ( x )= minimum ( x ^ n n ! , (1- x )^ n n ! ) , for 0 x 1 . Let I _ n = ₀¹ f _ n ( x ) dx , n 1 . Then I_ n = _ n=1 ^ I_ n is equal to-

Options

Correct answer

Step-by-step solution

More from Definite Integration