The roots of x2+ x + 12= 0 are a = −(1 + i)/2 and b = a

Problem 11876

(American Mathematical Monthly, Vol.122, December 2015) Proposed by A. Cibulis (Latvia).

Leta and b be the roots of x²+ x + ¹₂ = 0. Find

∞

X

n=1

(−1)ⁿ(aⁿ+ bⁿ) n + 2 .

Solution proposed by Roberto Tauraso, Dipartimento di Matematica, Universit`a di Roma “Tor Vergata”, via della Ricerca Scientifica, 00133 Roma, Italy.

The roots of x²+ x + ¹₂= 0 are a = −(1 + i)/2 and b = a.

Moreover for any complex number z 6∈ {−1, 0} such that |z| ≤ 1,

∞

X

n=1

(−z)ⁿ n + 2 = 1

z²

∞

X

n=1

(−1)ⁿzⁿ⁺²

n + 2 =−(ln(1 + z) − z + z²/2)

z² = −ln(1 + z) z² +1

z−1 2.

Hence

∞

X

n=1

(−1)ⁿ(aⁿ+ bⁿ)

n + 2 =

∞

X

n=1

(−1)ⁿ(aⁿ+ aⁿ) n + 2 = 2Re

∞

X

n=1

(−a)ⁿ n + 2

!

= 2 Re



−ln(1 + a) a² +1

a−1 2



= π − 3.