Determine the value of a &gt; 1 such that <munder> <mo movablelimits="true" form="pr

uplakanimkk

uplakanimkk

Answered question

2022-07-13

Determine the value of a > 1 such that
lim n → ∞ 1 + 2 3 + 3 3 + ⋯ + n 3 n 7 3 ( 1 ( a n + 1 ) 2 + 1 ( a n + 2 ) 2 + 1 ( a n + 3 ) 2 + ⋯ + 1 ( a n + n ) 2 ) = 54

Answer & Explanation

Allison Pena

Allison Pena

Beginner2022-07-14Added 14 answers

Considering
A n = ∑ k = 1 n k 3 n 7 3 ∑ k = 1 n 1 ( a n + k ) 2 
∑ k = 1 n k 3 = H n ( − 1 3 ) 
∑ k = 1 n 1 ( a n + k ) 2 = ψ ( 1 ) ( a n + 1 ) − ψ ( 1 ) ( a n + n + 1 )
A n = H n ( − 1 3 ) n 7 3 [ ψ ( 1 ) ( a n + 1 ) − ψ ( 1 ) ( a n + n + 1 ) ] 
Using asymptotics now for big n values (remembering that a > 1
H p ( − 1 3 ) = 3 p 4 / 3 4 + p 1 / 3 2 + ζ ( − 1 3 ) + O ( 1 p 2 / 3 ) 
ψ ( 1 ) ( p ) = 1 p + 1 2 p 2 + O ( 1 p 3 ) 
To make, use the last two times and keep going with the Taylor series.
n 7 3 ∑ k = 1 n 1 ( a n + k ) 2 = n 4 / 3 a 2 + a + 1 2 ( 1 ( a + 1 ) 2 − 1 a 2 ) n 1 / 3 + O ( 1 n 5 / 3 ) The situation is now straightforward, and you have both the limit and asymptotics.

dream13rxs

dream13rxs

Beginner2022-07-15Added 4 answers

I can clearly tie the remaining limit to a specific integral by consulting issues regarding limitations on MSE.
lim n → ∞ n ∑ k = 1 n 1 ( a n + k ) 2 = lim n → ∞ 1 n ∑ k = 1 n 1 ( a + k n ) 2 = ∫ 0 1 d x ( a + x ) 2 = 1 a 2 + a 
It follows that
a 2 + a = 72  ⟹  a 2 + a − 72 = ( a + 9 ) ( a − 8 ) = 0  ⟹  a = 8 

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