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Leah Pope 2022-06-29 Answered
Show that lim n [ n n + 1 + ( n + 1 ) n n n + 1 ] n = e e . All we have done was elementary manipulations, but we got stuck.
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Answers (1)

lilao8x
Answered 2022-06-30 Author has 22 answers
Define u n = ( 1 + 1 n ) n
( 1 + 1 n ( 1 + 1 n ) n ) n
= ( 1 + 1 n u n ) n
= exp ( n log ( 1 + u n n ) )
= exp ( u n + O ( 1 n ) ) The O ( n 1 ) term drops to 0 and u n e in the limit, so the value sought is exp ( e ) = e e
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