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auto23652im 2022-07-12 Answered
Consider the sum
S n = r = 2 n ( 1 ) r ( n r ) r 1
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Answers (1)

karburitc
Answered 2022-07-13 Author has 7 answers
Note that
S n = r = 2 n ( 1 ) r r 1 ( n r ) = n r = 1 n 1 ( 1 ) r + 1 r ( r + 1 ) ( n 1 r ) = n r = 1 n 1 ( 1 ) r + 1 r ( n 1 r ) + n r = 1 n 1 ( 1 ) r r + 1 ( n 1 r ) = n H n 1 + n 0 1 [ ( 1 x ) n 1 1 ] d x = n H n 1 n n 1 n = n H n n .
Thus, we have an exact expression for S n in terms of H n . The asymptotic expansion follows from that of H n , i.e.,
S n n log n ( 1 γ ) n + 1 2 1 12 n + 1 120 n 3
as n +

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