Find a closed form of a n </msub> similar to the following : <munderover> &#x

dglennuo

dglennuo

Answered question

2022-05-18

Find a closed form of a n similar to the following :
r = 0 n ( n r ) x r = ( 1 + x ) n

Answer & Explanation

extractumzz

extractumzz

Beginner2022-05-19Added 9 answers

a n = r = 1 n ( n r ) f ( r 1 ) ( 1 ) ( r 1 ) !
Note that:
Therefore,
a n = n r = 1 n ( n 1 r 1 ) f ( r 1 ) ( 1 ) r !
n ! a n = n r = 1 n ( n 1 r 1 ) ( n ! r ! ) f ( r 1 ) ( 1 )
= n r = 1 n ( n 1 r 1 ) ( d ( n r ) d x ( n r ) x n ) f ( r 1 ) ( x ) | x = 1
= n r = 0 n 1 ( n 1 r ) ( d ( n 1 r ) d x ( n 1 r ) x n ) f ( r ) ( x ) | x = 1
n ! a n = n d ( n 1 ) d x ( n 1 ) ( x n f ( x ) ) | x = 1
by Leibniz rule. Hence:
a n = 1 ( n 1 ) ! d ( n 1 ) d x ( n 1 ) ( x n f ( x ) ) | x = 1

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