For n ≥ 1, consider the sequence u n 3 = − n ( u n + 1 ). Evaluate lim n → ∞ u n

You know that ( u n ) is monotonic and bounded. So it has a limit. If u n → u ≠ − 1 then u n 3 = − n ( u n + 1 ) tends to ± ∞ (depending on whether u + 1 &gt; 0 or u + 1 &lt; 0) which is a contradiction. Hence, lim u n = − 1

Get the solution to "For n ≥ 1, consider the sequence u..."

shelohz0

Answered question

2022-05-18

For

n \geq 1

, consider the sequence

u_{n}^{3} = - n (u_{n} + 1)

. Evaluate

lim_{n \to \infty} u_{n}

Answer & Explanation

aqueritztv

Beginner2022-05-19Added 10 answers

You know that

(u_{n})

is monotonic and bounded. So it has a limit. If

u_{n} \to u \neq - 1

then

u_{n}^{3} = - n (u_{n} + 1)

tends to

\pm \infty

(depending on whether

u + 1 > 0

u + 1 < 0

) which is a contradiction. Hence,

lim u_{n} = - 1

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