MathDB

Define in the plane the sequence of vectors

v_1, v_2, \ldots

with initial values

v_1 = (1, 0)

v_2 = (-1/\sqrt{2}, 1/\sqrt{2})

and satisfying the relationship

v_n=\frac{v_{n-1}+v_{n-2}}{\lVert v_{n-1}+v_{n-2}\rVert},

for

n \geq 3

. Show that the sequence is convergent and determine its limit.
Note: The expression

\lVert v \rVert

denotes the length of the vector

v

Problem Statement