Derive generalized convergence point for this infinite series at a given angle.
Answers
Given : bottom-most sheet has width 1 unit and length √2 unit and its bottom-left corner is the origin
To Find : coordinates of the point at which the A series shown in the image below would converge
generalized convergence point for this infinite series at a given angle
Solution:
Bottom sheet width = 1 unit
and length/height = √2
Size of further each sheet has length & width divided by √2
hence r = 1/√2
Width GP
a = 1 , r = 1/√2
Sum of infinite GP = a/(1 - r)
= 1/ ( 1 - 1/√2 )
= √2/(√2 - 1)
= √2(√2 + 1) ( by rationalizing )
= 2 + √2
Length/height GP
a = √2 , r = 1/√2
Sum of infinite GP = a/(1 - r)
= √2/ ( 1 - 1/√2 )
= 2/(√2 - 1)
= 2(√2 + 1) ( by rationalizing )
= 2√2+ 2
Hence Coordinates of the point at which Series converges
( 2 + √2 , 2√2+ 2 )
generalized convergence point for this infinite series at a given angle α
( ( 2 + √2 )Cos α - (2√2+ 2 ) Sin α , ( 2 + √2 )Sin α + (2√2+ 2 )Cos α )
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