A NOTE ON THE SUM ∑ 1/w_{k2^{n}}

Stanley Rabinowitz

1. Historical Results. In 1974, Millin [13] published a problem stating that

∑_{n=0}^{∞} \frac{1}{F_{2^{n}}} = \frac{7 − \sqrt{5}}{2}.  (1)

This spurred a flurry of activity: [1, 3, 4, 5, 6, 7, 8, 17]. Most investigators, however, overlooked the fact that Lucas studied such sums back in 1878. He showed in [11], equation (125), that if k ≠ 0, then

∑_{n=1}^{N} \frac{Q^{k2^{n−1}}}{u_{k2^{n}}} = \frac{Q^{k}u_{k(2^{N} −1)}}{u_{k}u_{k2^{N}}}  (2)

where u_{n} is a second order linear recurrence defined by

u_{n+2} = P u_{n+1} − Qu_{n}, u_{0} = 0, u_{1} = 1.

If we use the identity Q^{n−1}u_{m−n} = u_{n}u_{m−1} −u_{m}u_{n−1}, we can express formula (2) in the form

∑_{n=1}^{N} \frac{Q^{k2^{n−1}}}{u_{k2^{n}}} = Q[\frac{u_{k2^{N} −1}}{u_{k2^{N}}} − \frac{u_{k−1}}{u_{k}} ].  (3)

If Q = −1, as is the case for Fibonacci, Lucas, and Pell numbers, then equation (3) becomes

∑_{n=0}^{N} \frac{1}{u_{k2^{n}}} = \frac{1 + u_{k−1}}{u_{k}} + \frac{1 − (−1)^{k}}{u_{2k}} − \frac{u_{k2^{N} −1}}{u_{k2^{N}}}  (4)

where we have handled the terms when n is 0 and 1 specially. For all subsequent terms, the exponent of Q is even and hence the numerator is 1. An equivalent formula found by Greig [6] is

∑_{n=0}^{N} \frac{1}{u_{k2^{n}}} = \frac{1}{u_{k}} + \frac{1 + u_{2k−1}}{u_{2k}} − \frac{u_{k2^{N} −1}}{u_{k2^{N}}} .  (5)

When < u_{n} > is the Fibonacci sequence, equation (4) becomes the result found by Greig in [5]. Hoggatt and Bicknell [8] found an equivalent result, expressing their answer in terms of Fibonacci and Lucas numbers. This generalized the result they gave in [7]. Brady [2] found an equivalent result, expressing his answer in terms of the golden ratio. When < u_{n} > is the Pell sequence, equation (4) becomes the result found by Horadam [10]. In equation (3), if we let Q = 1, we get the results found by Melham and Shannon [12].

Lucas [11] also found that if k ≠ 0 and p ≠ 0, then

∑_{n=0}^{N} \frac{Q^{kp^{n}}u_{k(p−1)p^{n}}}{u_{kp^{n}}u_{kp^{n+1}}} = \frac{Q^{k}u_{k(p^{N+1}−1)}}{u_{k}u_{kp^{N+1}}} .  (6)

This, again, was overlooked by later researchers. Formula (6) is equivalent to equation (6) of Bruckman and Good [3]. If we let P = x and Q = −1, then we get a result found by Popov [16], equation (4), for the Fibonacci polynomials. This, in turn, generalizes results for Fibonacci numbers found by Bergum and Hoggatt [1]. Brady [2] found an equivalent result for Fibonacci numbers, expressing his answer in terms of the golden ratio.

2. New Results. Instead of the sequence < u_{n} >, we can study the sequence < w_{n} > defined by

w_{n+2} = P w_{n+1} − Qw_{n}, w_{0}, w_{1} arbitrary.

In order that no denominator be 0, we will make the assumption that w_{n} ≠ 0 for n > 0. We also assume that k is a fixed positive integer and that P^{2} ≠ 4Q. Finally, we let

α = \frac{P + \sqrt{P^{2} − 4Q}}{2} and β = \frac{P − \sqrt{P^{2} − 4Q}}{2}

and note that αβ = Q.

In [10], a formula for ∑ 1/w_{k2^{n}} is claimed to be found for the case where Q = −1. However, this formula is not correct unless w_{0} = 0. For k = 1, the supposed formula is

∑_{n=0}^{N} \frac{1}{w_{2^{n}}} = \frac{1}{w_{1}} + \frac{1 + w_{1}}{w_{2}} − \frac{w_{2^{N} −1}}{w_{2^{N}}} .

A counterexample to this claim is the Lucas sequence with N = 2. Perhaps the author inadvertently omitted the hypothesis w_{0} = 0, in which case the above formula and the formulas given on page 112 of [10] are valid. These results are then a special case of the following.

\underline{Theorem 1}. If w_{0} = 0, then

∑_{n=1}^{N} \frac{Q^{k2^{n−1}}}{w_{k2^{n}}} = \frac{Q^{k}w_{k(2^{N} −1)}}{w_{k}w_{k2^{N}}} .  (7)

\underline{Proof}. We use the identity w_{n} = w_{1}u_{n} − Qw_{0}u_{n−1} which comes from [9]. Letting w_{0} = 0, we find that w_{n} = w_{1}u_{n} for all n. Substituting u_{n} = w_{n}/w_{1} in equation (2) gives us the desired result.

\underline{Corollary 1}. If w_{0} = 0 and Q = 1, then

∑_{n=1}^{N} \frac{1}{w_{k2^{n}}} = \frac{w_{k(2^{N} −1)}}{w_{k}w_{k2^{N}}} = w_{1}[\frac{w_{k2^{N} −1}}{w_{k2^{N}}} − \frac{w_{k−1}}{w_{k}} ].  (8)

\underline{Corollary 2}. If w_{0} = 0 and Q = −1, then

∑_{n=1}^{N} \frac{1}{w_{k2^{n}}} = \frac{1 − (−1)^{k}}{w_{2k}} + \frac{w_{k(2^{N} −1)}}{w_{k}w_{k2^{N}}} = \frac{1 + w_{2k−1}}{w_{2k}} − \frac{w_{k2^{N} −1}}{w_{k2^{N}}} .  (9)

In a similar manner, formula (6) continues to hold when u is replaced by w, provided that w_{0} = 0.

Sums to infinity can also be obtained by letting N → ∞ in any of the above formulas. We use the following fact, which is taken from [15].

\underline{Lemma}. For all integers r,

lim_{N→∞} \frac{u_{N−r}}{u_{N}} = \{ \begin{matrix}α^{r}, & if |β/α| < 1, \\ β^{r}, & if |β/α| > 1. \end{matrix}

When w_{0} = 0, so that w_{n} is proportional to u_{n}, we may replace u by w in the above lemma. Letting N → ∞ in formula (7) and recalling that αβ = Q, we get the following.

\underline{Theorem 2}. If w_{0} = 0, then

∑_{n=1}^{∞} \frac{Q^{k2^{n−1}}}{w_{k2^{n}}} = \{ \begin{matrix}β^{k}/w_{k}, & if |β/α| < 1, \\ α^{k}/w_{k}, & if |β/α| > 1. \end{matrix}  (10)

If < w_{n} > is the Fibonacci sequence, then formula (10) reduces to formula (1), and this agrees with the value found by Lucas in 1878: formula (127) of [11].

\underline{References}

1. G. E. Bergum and V. E. Hoggatt, Jr., “Infinite Series with Fibonacci and Lucas Polynomials,” The Fibonacci Quarterly, 17 (1979), 147–151.

2. W. G. Brady, “Additions to the Summation of Reciprocal Fibonacci and Lucas Series,” The Fibonacci Quarterly, 9 (1971), 402–404, 412.

3. P. S. Bruckman and I. J. Good, “A Generalization of a Series of De Morgan with Applications of Fibonacci Type,” The Fibonacci Quarterly, 14 (1976), 193–196.

4. I. J. Good, “A Reciprocal Series of Fibonacci Numbers,” The Fibonacci Quarterly, 12 (1974), 346.

5. W. E. Greig, “Sums of Fibonacci Reciprocals,” The Fibonacci Quarterly, 15 (1977), 46–48.

6. W. E. Greig, “On Sums of Fibonacci–Type Reciprocals,” The Fibonacci Quarterly, 15 (1977), 356–358.

7. V. E. Hoggatt, Jr. and M. Bicknell, “A Primer for the Fibonacci Numbers, Part XV: Variations on Summing a Series of Reciprocals of Fibonacci Numbers,” The Fibonacci Quarterly, 14 (1976), 272–276.

8. V. E. Hoggatt, Jr. and M. Bicknell, “A Reciprocal Series of Fibonacci Numbers with Subscripts 2^{n}k,” The Fibonacci Quarterly, 14 (1976), 453–455.

9. A. F. Horadam, “Basic Properties of a Certain Generalized Sequence of Numbers,” The Fibonacci Quarterly, 3 (1965), 161–176.

10. A. F. Horadam, “Elliptic Functions and Lambert Series in the Summation of Reciprocals in Certain Recurrence–Generated Sequences,” The Fibonacci Quarterly, 26 (1988), 98–114.

11. E. Lucas, “Théorie des Fonctions Numériques Simplement Périodiques,” American Journal of Mathematics, 1 (1878), 184–240, 289–321.

12. R. S. Melham and A. G. Shannon, “On Reciprocal Sums of Chebyshev Related Sequences,” The Fibonacci Quarterly, 33 (1995), 194–202.

13. D. A. Millin, “Problem H–237,” The Fibonacci Quarterly, 12 (1974), 309.

14. B. S. Popov, “On Certain Series of Reciprocals of Fibonacci Numbers,” The Fibonacci Quarterly, 22 (1984), 261–265.

15. B. S. Popov, “Summation of Reciprocal Series of Numerical Functions of Second Order,” The Fibonacci Quarterly, 24 (1986), 17–21.

16. B. S. Popov, “A Note on the Sums of Fibonacci and Lucas Polynomials,” The Fibonacci Quarterly, 23 (1985), 238–239.

17. A. G. Shannon, “Solution to Problem H–237: Sum Reciprocal!,” The Fibonacci Quarterly, 14 (1976), 186–187.

Stanley Rabinowitz MathPro Press 12 Vine Brook Road Westford, MA 01886 email: stan@wwa.com

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