This summer I set out to create a set of interesting integrals involving infinite nested radicals. In so doing, I went on to derive a number of closed formulae for infinite nested radicals with periodic base and root sequences. This section is thus devoted to describing what I’ve learned through this exercise.
Nested Radicals as Infinite Products
Let be a periodic sequence of positive terms with period so that for all . Let us consider the infinite nested radical
Provided converges, we find from periodicity of that
so that rearranging gives
But observe that the expression for is formally equivalent to
Thus, we find our sequence satisfies
Where the convergence of is concerned, this is guaranteed by the following result whenever the sum converges (see Infinite Product on Wolfram MathWorld).
Theorem 1. Given a sequence of positive numbers , the product
converges to a nonzero number if and only if the following series converges:
Now, notice that any periodic sequence of real numbers is necessarily bounded. In particular, if is a periodic sequence of positive numbers with period , we see that , so that is always finite. Consequently, so is the quantity . But then,
Hence, the sum converges absolutely and, as such, the product always converges by Theorem 1. With the formal equivalence
in mind, we conclude the following.
Corollary 1. Let be a periodic sequence of positive numbers with period so that for all . Then
converges and there holds
Example 1. Consider the sequence given by and for with some and given. Corollary 1 allows us to explicitly determine given a period . Let us write to indicate the dependence of on . We then have
from which we calculate
Notably, in the infinite period limit as we find
Alternatively, the same result can be deduced directly by considering without imposing any periodicity. As such we find
which gives the same result after evaluating the infinite sum in the exponent. Note that in general,
if and .
Our first nested radical
leads us to consider the same calculation for the general nested radical
where and are periodic sequences of positive numbers in-phase, which means there exists some such that and for all . Let’s say that the sequence is the root sequence and is the base sequence in the above expression. In this case, we have a generalisation of Corollary 1 whose assumptions allow us to deduce immediately that the sum
converges absolutely. Our approach to proving Corollary 1 thus applies similarly, giving
Theorem 2. Let and be periodic sequences of positive numbers in phase with period . Suppose . Then
converges. Moreover, if then
while if then
Thus far, our analysis concerned periodic base sequences that are in phase with periodic root sequences. What can be said when
- Either sequence , is not periodic, or
- Both sequences and are periodic but out of phase?
The following example considers an extreme case where isn’t periodic while is periodic.
Example 2. Suppose is a periodic sequence with period so that for , and let be the sequence given by , . Then, we have the expression
converges because the sum converges absolutely by periodicity of . We then find
There appears to be no closed form for while it is clear that the sum converges. With help from Wolfram MathWorld, we may express in terms of the generalized hypergeometric function whereby
In the case where , so that is a constant sequence, we find
On the other hand, we have an example of a nested radical with periodic root sequence and non-periodic base sequence .
Example 3. Take , for . We then consider the expression
By Theorem 1 we see that the above infinite product converges because the following sum converges (by the comparison test):
Then, with help from Wolfram Alpha, we find that
Remark 1. We see from our two extreme examples that familiarity with special functions can be beneficial to arriving at closed forms for out-of-phase periodic nested infinite radicals.
Now, suppose and are periodic base and root sequences with respective periods and , whereby and for . If , for some we have the following result that allows for exact computation of the corresponding nested infinite radical.
Theorem 3. Let and be periodic sequences of real numbers with periods such that , and for all while for some . If , then
Remark 2. Notice from the conditions of Theorem 3 that we can assign to each an arbitrary sign, as long as for all . With this flexibility, we’ve managed to cover other types of infinite nested radicals such as
When the divides , we have the following theorem whose result generalises Theorem 2.
Theorem 4. Let and be periodic sequences of real numbers with periods which satisfy for some while , and for all . If then
From Theorem 3 we can deduce the case where is an affine function of , in that for some with and . As such, the following result generalises Theorem 4.
Theorem 5. Let and be periodic sequences of real numbers with periods such that , and for all , while
for some where and .
Suppose . Then
Theorem 5 covers the evaluation of our nested radicals when the base and root periods are related via . So, what can be said when the periods are swapped, i.e when ? It turns out that all of our previous results are contained in the following in which no particular relationship between and needs to be assumed.
Theorem 6. Let and be periodic sequences of real numbers with periods such that , and for all . If . Then
for with and .
Theorem 6 allows us to determine some nested radicals with periodic sequences having arbitrary periods. In particular, if we hold fixed, sending to infinity appropriately gives a formula for calculating our general nested radical whenever the root sequence
- isn’t necessarily periodic,
- satisfies for all , and
- contains a subsequence for which for all , with some , and the sequence has polynomial growth.
Assuming this, consider a periodic approximation of with period . Then, we essentially find that , and in the limit as is sent to infinity in such a way that divides (since then in this limit). Consequently, we have for each ,
which implies that
for . Note that the polynomial growth assumption guarantees the series converges absolutely for each . To summarise, we have
Corollary 2. Let be a periodic sequence of positive numbers with period so that for . Suppose is a sequence of real numbers for which for all and it contains a subsequence which satisfies for all , with some , and the sequence has polynomial growth. Then
where the sum
converges for each .
As another corollary of Theorem 6 we derive a result similar to Theorem 4 but with the condition instead of .
Corollary 3. Let and be periodic sequences of real numbers with periods such that , and for all and for some . Suppose . Then
Remark 3. Notice the interesting difference between the formulae of in Theorem 4 and Corollary 3.
- Theorem 4:
- Corollary 3:
By comparing the two formulae, we see order 1 dependence on the multiplier as tends to infinity, although the role played by this multiplier is different between the cases. In fact, taking to infinity we find
- Theorem 4: while finite and
- Corollary 3: while finite and
Example 4. With the formula given by Theorem 6, we can derive a curious formula for the determinant of the matrix with entries given by , for some positive numbers , and as given in the theorem. In particular, set and, given a permutation , let our periodic base sequence be given by for , leaving the root sequence unspecified (though satisfying the assumptions of Theorem 6). Then, we have
The definition of determinant of thus gives
We have thus concluded our analysis of infinite nested radicals that are given by infinite products. Indeed, we have not covered explicit evaluation of all such nested radicals. As such, the reader is now invited to answer the following question.
Open Question. Are there closed forms for the following expressions?
Copyright © 2020 Yohance Osborne