Don't find the exact formula

Calculus Level 3

Define sequences ( a n ) n ∈ N (a_n)_{n\in \mathbb{N}} and ( b n ) n ∈ N (b_n)_{n\in \mathbb{N}} as follows:

  • a 1 = 0 a_1 = 0
  • b 1 = 1 b_1 = 1
  • a n + 1 = a n + 2 b n a_{n+1} = a_n + 2b_n
  • b n + 1 = a n + b n b_{n+1} = a_n + b_n

Find lim ⁡ n → ∞ a n b n \displaystyle \lim_{n\to\infty} \frac{a_n}{b_n} to 3 decimal places.


The answer is 1.414214.

Jonathan Dunay by Jonathan Dunay , 22, USA

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1 solution

Jonathan Dunay
Mar 26, 2017

Define ( y n ) n i n N (y_n)_{n \ in \mathbb{N}} as y n = a n b n y_n = \frac{a_n}{b_n} . Then, using the equations, we can say that y n + 1 = y n + 2 y n + 1 y_{n+1} = \frac{y_n + 2}{y_n + 1} . If the limit of both sides of the equation exists, then they must be equal. We are already assuming that lim ⁡ y n \lim y_n exists (call this limit L L ). So, taking the limit of the equation, we get that L = L + 2 L + 1 L = \frac{L + 2}{L + 1} . Solving we get that L = ± 2 L = \pm \sqrt{2} . Since the sequence is always positive we have that L = 2 ≈ 1.414214 … L = \sqrt{2} \approx 1.414214\ldots

For bonus, prove that the sequence converges.

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