The limit of a sequence Given x_n=(1)/(n^2+1)+(1)/(n^2+2)+(1)/(n^2+3)+...+(1)/(n^2+n) Verify if there is or no a limit. Find it if affirmative. Let a_n=(n)/(n^2+1) (The biggest portion of the sum n times) and b_n=(n)/(n^2+n) (the smallest portion of the sum n times) then b_n<=x_n<=a_n since lim (n)/(n^2+1)=lim (n)/(n^2+n)=0, we have that lim x_n=0. Is this wrong? why? If it is, any tips on how to find limxn? Grateful for any help.

Ledexadvanips 2022-08-12 Answered
The limit of a sequence
Given
x n = 1 n 2 + 1 + 1 n 2 + 2 + 1 n 2 + 3 + + 1 n 2 + n
Verify if there is or no a limit. Find it if affirmative.
Let a n = n n 2 + 1 (The biggest portion of the sum n times) and b n = n n 2 + n (the smallest portion of the sum n times) then
b n x n a n
b n x n a n
since
lim n n 2 + 1 = lim n n 2 + n = 0 ,
we have that
lim x n = 0.
Is this wrong? why? If it is, any tips on how to find lim x n ? Grateful for any help.
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Answers (1)

neglegir86
Answered 2022-08-13 Author has 12 answers
Your reasoning is true, but I think the following a bit of better.
0 < x n < n 1 n 2 = 1 n
Thus,
0 lim n + x n lim n + 1 n = 0 ,
which says lim n + x n = 0
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