Evaluating lim_(n => infty) (1-(x)/(n^(1+a)))^n

Garrett Sheppard 2022-08-14 Answered
Evaluating lim n ( 1 x n 1 + a ) n
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Answers (1)

Brennan Parks
Answered 2022-08-15 Author has 14 answers
That step you are uncomfortable with is indeed quite dodgy. Instead, consider the logarithm of your limit. Since log ( 1 t ) = t + t 2 / 2 + t 3 / 3 + ,
n log ( 1 x n 1 + a ) = ( x n a + x 2 2 n 1 + 2 a + x 3 3 n 2 + 3 a + )
so as n , if a = 0 this goes to x, if a > 0 to goes to 0 and if a < 0 this goes to .. Exponentiating gives back the result.
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