Question

8. Derive generalized convergence point for this infinite series at a given angle.​

Answer 1

Answer:

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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