Does this sequence x_n=(sqrt[n](e)-1)* n converge?

bucstar11n0h

bucstar11n0h

Answered question

2022-11-07

Does the sequence defined by
x n = ( e n 1 ) n converge?

Answer & Explanation

dobradisamgn

dobradisamgn

Beginner2022-11-08Added 17 answers

You have:
x n = ( e n 1 ) n
Expanding e n in series will give:
1 + 1 n + 1 2 n 2 + O ( ( 1 n ) 3 )
So
( n e 1 ) n = ( 1 n + 1 2 n 2 + O ( ( 1 n ) 3 ) ) n = 1 + 1 2 n + O ( ( 1 n ) 3 ) n
and
lim x x n = 1

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