How to calculate &#x222B;<!-- ∫ --> <mrow class="MJX-TeXAtom-ORD"> Q </

Jasmin Pineda

Jasmin Pineda

Answered question

2022-06-16

How to calculate
Q ( x 1 + x 2 + + x n ) 2 d λ n
whereas n 2 and
Q = { ( x 1 , , x n ) R n : 0 x i 1 , i = 1 , , n }

I know that I have to use induction and Fubini to calculate
I ( n ) = 0 1 0 1 0 1 ( x 1 + x 2 + + x n ) 2 d x n d x 2 d x 1
And I already started to calculate the integrals for n = 2 , 3 , 4 but the numbers I got did not enlighten me and I have no idea for an induction hypothesis. I also exchanged x with 1 but I still have no clue. Maybe I am doing something wrong. Thanks for help in advance.

Answer & Explanation

Blaze Frank

Blaze Frank

Beginner2022-06-17Added 18 answers

By expanding
( x 1 + + x n ) 2 = i = 1 n x i 2 + 2 1 i < j n x i x j
we see that
I ( n ) = i = 1 n 1 3 + 2 1 i < j n 1 2 1 2 = n 3 + 1 2 n ( n 1 ) 2 = n ( 3 n + 1 ) 12
Alternatively, use the identity
( x 1 + + x n ) 2 = ( x 1 + + x n 1 ) 2 + 2 x n ( x 1 + + x n 1 ) + x n 2
to obtain the recursion
I ( n ) = I ( n 1 ) + n 1 2 + 1 3
Since I ( 1 ) = 1 / 3, then
I ( n ) = 1 3 + j = 1 n 1 ( j 2 + 1 3 )
which evaluates to n ( 3 n + 1 ) / 12.

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