Iterated Roots Convergence

Let the sequence be $x_n = \sqrt{ 1 + \sqrt{ 2 + \ldots + \sqrt{n } } }$ . Clearly $x_n$ is a monotonically increasing seqeunce. Thus, we just need to show that they have an upper bound, which shows that this sequence converges.

Claim: For all $n$ , $x_n \leq 2$ .

Proof: By squaring all of the roots, we ultimately want to show that

$P_n: (((( 2^2 - 1 ) ^2 - 2 ) ^ 2 - 3 ) ^ 2 \ldots )^2 - n > 0$

However, note that if this is true, that it doesn't yet allow us to conclude anything about $P_{n+1}$ . Thus, we want to show something stronger , which allows us to proceed via induction. By playing around, we see that if

$(((( 2^2 - 1 ) ^2 - 2 ) ^ 2 - 3 ) ^ 2 \ldots )^2 - n > \sqrt{ 3(n+1) } .$

is true for some $n$ , we can then conclude that

$((((( 2^2 - 1 ) ^2 - 2 ) ^ 2 - 3 ) ^ 2 \ldots )^2 - n)^2 - (n+1) > 3(n+1) - (n+1) = 2 (n+1) > \sqrt{3 (n+2) }$

This allows us to complete the induction step! Of course, there could be numerous possibilities for the RHS, like $2 \sqrt{n+1}$ or \sqrt{2 (n+3) . I just chose this one first.

All that remains is to find a large enough base case where the inequality is true. In fact, this is already satisfied at $n = 1$ since

$2^2 - 1 = 3 > \sqrt{ 3 \times 2} .$

Hence, we can proceed by induction.

We can compare the sequence $a_n$ with another sequence $b_n = \left\lceil \sqrt{1 + \left\lceil \sqrt{2 + \left \lceil \sqrt{3 + \cdots + \left\lceil \sqrt{n} \right \rceil } \right\rceil}\right\rceil } \right\rceil$ . We see that $b_n \ge a_n$ for all $n$ .

We see that $b_1 = 1$ . We can show that $b_n = 2$ for all $n \ge 2$ .

Since $a_n$ is strictly increasing sequence, and it has an upper bound, it converges.

This is the famous "Nested Radical Constant" with extensive literature on the subject. In particular, Vijayaraghavan proved that the sequence

$\sqrt { { a }_{ 1 }+\sqrt { { a }_{ 2 }+\sqrt { { a }_{ 3 } +\sqrt { { a }_{ 4 }\;+... } } } }$

for positive ${a}_{n}$ converges if and only if the Limit superior of $\dfrac { Log\left( { a }_{ n } \right) }{ { 2 }^{ n } }$ is less than infinity.

There is no known closed form expression for this Nested Radical Constant.