# Convergence of $a_1=1,a_2=\sqrt{7},a_3=\sqrt{7\sqrt{7}}, a_4=\sqrt{7\sqrt{7\sqrt{7}}}, \dots$

Define a sequence $a_n$ as follows: $a_1=1,a_2=\sqrt{7},a_3=\sqrt{7\sqrt{7}}, a_4=\sqrt{7\sqrt{7\sqrt{7}}}, \dots$

Determine if it's convergent and find its limit.

The sequence satisfies $a_n=\sqrt{7a_{n-1}}$. If it's convergent with limit $a$, then $a^2=7a$, so either $a=0$ or $a=7$. We show that the sequence is monotonically increasing and bounded above by $7$ so its limit cannot be equal $0$, thus it is $7$.

Claim. $a_n < 7$ for all $n$.

Proof: Induction on $n$. The base is clear. Assume $a_{n-1} < 7$. This is equivalent to saying that $7a_{n-1} < 49$, which happens iff $\sqrt {7a_{n-1}} < 7$. Then $a_n=\sqrt{7a_{n-1}} < 7$.

It remains to show $a_n$ is monotonically increasing. Consider $a_n/a_{n-1}=\sqrt{7/a_{n-1}}$. We prove that this is greater than $1$. It will follow that $a_n > a_{n-1}$.

Since $a_n < 7$, $7/a_{n-1}> 1$, whence $a_n/a_{n-1} > 1$. Thus the sequence is increasing and bounded above, so it's convergent. It's limit cannot be zero, so it must be $7$.

Is this a correct proof?

• math.stackexchange.com/questions/589288/… – lab bhattacharjee Aug 9 '18 at 18:00
• Although that question contains a complete solution, it doesn't answer whether my reasoning is correct. – user531232 Aug 9 '18 at 18:04
• It's simpler, just note that $a_n = 7^{\frac12+\frac14+\frac18+...+2^{1-n}}$. Now just sum the geometric series and use continuity of $x \mapsto 7^x$. – Shalop Aug 9 '18 at 18:14
• @Shalop For me it's not simpler, actually. I was aware of that solution, but the solutions that I produce on my own tend to be of more help, so I posted this to check whether my version has any mistakes. – user531232 Aug 9 '18 at 18:19
• Yes, the proof is correct. – saulspatz Aug 9 '18 at 18:33

It all boils down to showing that $$0<a_n<7\implies a_n<\sqrt{7a_n}=a_{n+1}<7,$$ which is fairly obvious (geometric average).
Let $T(a)=\sqrt{7a}$. For $a \geq 2$, $$T(a) \geq \sqrt{14} > \sqrt{4} = 2 \qquad \text{and} \qquad |T^{\prime}(a)| = \frac{\sqrt{7}}{2\sqrt{a}} \leq \frac{\sqrt{7}}{2\sqrt{2}} = \frac{\sqrt{14}}{4} < \frac{\sqrt{16}}{4} = 1.$$ Therefore, $T$ is a contraction on $[2, \infty)$.
By the Banach fixed point theorem, $T$ has a fixed point in $X$. You already proved that this fixed point has to be $a=7$ (since $a=0 \notin X$). Conclude by noting that $T(a_1)=T(1)\geq2$.