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a_n= a_{n-1} +1/n (cos 45^o + i sin 45^o )^n

Source: Spanish Mathematical Olympiad 1973 P3

December 23, 2022

complex analysisrecurrence relationRecurrenceSequenceanalysis

Problem Statement

The sequence

(a_n)

of complex numbers is considered in the complex plane, in which is:

a_0 = 1, \,\,\, a_n = a_{n-1} +\frac{1}{n}(\cos 45^o + i \sin 45^o )^n.

Prove that the sequence of the real parts of the terms of

(a_n)

is convergent and its limit is a number between

0.85

and

1.15