Find lim n → ∞ 1 n ( n + n − 1 2 +...

vrotterigzl

vrotterigzl

Answered

2022-06-24

Find lim n 1 n ( n + n 1 2 + n 2 3 + + 2 n 1 + 1 n log ( n ! ) )

Answer & Explanation

Trey Ross

Trey Ross

Expert

2022-06-25Added 30 answers

Let x n be the sequence given by
x n = 1 n ( k = 1 n n + 1 k k log ( k ) ) = 1 n ( k = 1 n n k + 1 k 1 log ( k ) ) (1) = 1 + 1 n k = 1 n 1 k 0 as n + k = 1 n 1 k log ( n ) γ as n 1 n k = 1 n log ( k / n ) 1 as n
Therefore, we find that
lim n x n = γ
Emmy Knox

Emmy Knox

Expert

2022-06-26Added 10 answers

By the Stolz-Cesàro theorem
lim n x n = lim n 1 n k = 1 n ( n + 1 k k log ( k ) ) = lim n k = 1 n ( n + 1 k k log ( k ) ) n = lim n k = 1 n + 1 ( n + 2 k k log ( k ) ) k = 1 n ( n + 1 k k log ( k ) ) ( n + 1 ) n = lim n k = 1 n + 1 ( n + 1 k k + 1 k log ( k ) ) k = 1 n ( n + 1 k k log ( k ) ) = lim n k = 1 n + 1 1 k + k = 1 n + 1 ( n + 1 k k log ( k ) ) k = 1 n ( n + 1 k k log ( k ) ) = lim n k = 1 n + 1 1 k log ( n + 1 ) = γ

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