Compute lim n → ∞ ( 3 n ) 1 / 3 x n

Lorena Lester

Lorena Lester

Answered

2022-07-22

Compute lim n ( 3 n ) 1 / 3 x n

Answer & Explanation

yelashwag8

yelashwag8

Expert

2022-07-23Added 17 answers

We have that
x n 0 , S n = k = 1 n x k 2
We use Stolz's lemma to show
lim n 3 n S n 3 = 1
And we would be done after that, indeed
lim n 3 n x n 3 = lim n 3 n x n 3 x n 3 S n 3 = lim n 3 n S n 3 = 1
and so by continuity of f ( x ) = x 1 3 we are done.
Now let's prove our claim
lim n 3 n S n 3 = lim n 3 ( n + 1 ) 3 n S n + 1 3 S n 3 =
= 3 ( S n + 1 S n ) ( S n + 1 2 + S n + 1 S n + S n 2 ) =
= 3 x n + 1 2 ( S n + 1 2 + S n + 1 S n + S n 2 ) =
= 3 x n + 1 2 S n + 1 2 ( 1 + S n S n + 1 + S n S n + 1 ) 2 =
Now
lim n S n S n + 1 = lim n ( 1 x n + 1 2 S n + 1 ) = 1
Because
x n 2 S n = x n 3 S n x n
So we are left
lim n 3 n S n 3 = lim n 3 x n + 1 2 S n + 1 2 1 3 = 1

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