a1 = √3 an+1 = √(3 + an) How do you prove that an is convergent? Would you find its limit by an tends to a (as n tends to inf.), so an+1 tends to a. Then solve for a by a2 = a + 3??
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Prove that $\displaystyle a_n<a_{n+1}<3$. Do this by induction. Then we know that an increasing bounded sequence converges. Then to find the limit solve this: $\displaystyle L=\sqrt{3+L}$
Originally Posted by Plato Prove that $\displaystyle a_n<a_{n+1}<3$. Do this by induction. Then we know that an increasing bounded sequence converges. Then to find the limit solve this: $\displaystyle L=\sqrt{3+L}$ thank you!! But how would you go about proving that by induction?
Originally Posted by pseudonym thank you!! But how would you go about proving that by induction? Well prove that $\displaystyle a_1 < a_2 <3$. Then show that $\displaystyle a_K<a_{K+1}<3$ implies $\displaystyle a_{K+1}<a_{K+2}<3$.
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