Maths-
General
Easy

Question

sum from n equals 0 to straight infinity of   open parentheses fraction numerator 2 i over denominator 3 end fraction close parentheses to the power of n equals

  1. fraction numerator 9 plus 6 i over denominator 13 end fraction
  2. fraction numerator 9 minus 6 i over denominator 13 end fraction
  3. 9+6i
  4. 9-6i

hintHint:

The sum of infinite GP is the sum of an infinite number of terms of a geometric progression (GP). We may think that the sum of an infinite number of terms is infinity always. But this is NOT true in the case of an infinite GP. The sum of infinite GP is a finite number when the absolute value of its common ratio is less than 1.

The correct answer is: fraction numerator 9 plus 6 i over denominator 13 end fraction


    sum from n equals 0 to straight infinity of   open parentheses fraction numerator 2 i over denominator 3 end fraction close parentheses to the power of n equals
    sum from n equals 0 to straight infinity of   open parentheses fraction numerator 2 i over denominator 3 end fraction close parentheses to the power of n equals   open parentheses fraction numerator 2 i over denominator 3 end fraction close parentheses to the power of 0 space plus space open parentheses fraction numerator 2 i over denominator 3 end fraction close parentheses to the power of 1 space plus space open parentheses fraction numerator 2 i over denominator 3 end fraction close parentheses squared plus space open parentheses fraction numerator 2 i over denominator 3 end fraction close parentheses cubed..............
space space space space space space space space space space space space space space space space space space equals space 1 space plus space open parentheses fraction numerator 2 i over denominator 3 end fraction close parentheses space plus space open parentheses fraction numerator 2 i over denominator 3 end fraction close parentheses squared plus space open parentheses fraction numerator 2 i over denominator 3 end fraction close parentheses cubed..............
W e space k n o w space t h e space semicolon space T h e space f o r m u l a space t o space f i n d space t h e space s u m space o f space i n f i n i t e space g e o m e t r i c space p r o g r e s s i o n i s space S subscript infinity equals fraction numerator a over denominator left parenthesis 1 – r right parenthesis end fraction comma
w h e r e space bold italic a space i s space t h e space f i r s t space t e r m space a n d space bold italic r space i s space t h e space c o m m o n space r a t i o.
H e r e comma space bold space bold italic r space i s space open parentheses fraction numerator 2 i over denominator 3 end fraction close parentheses space t h e space c o m m o n space r a t i o space A n d space bold italic a space i s space 1 space t h e space f i r s t space t e r m...
s o space comma sum from n equals 0 to straight infinity of   open parentheses fraction numerator 2 i over denominator 3 end fraction close parentheses to the power of n space equals fraction numerator 1 over denominator left parenthesis 1 – open parentheses fraction numerator 2 i over denominator 3 end fraction close parentheses right parenthesis end fraction space equals fraction numerator 3 over denominator left parenthesis 3 – 2 i right parenthesis end fraction cross times fraction numerator left parenthesis 3 plus 2 i right parenthesis over denominator left parenthesis 3 plus 2 i right parenthesis end fraction
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space fraction numerator left parenthesis 9 plus 6 i right parenthesis over denominator 9 plus 4 end fraction space space space space space left curly bracket space W e space k n o w space t h a t space semicolon space i squared equals negative 1 space right curly bracket
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals fraction numerator left parenthesis 9 plus 6 i right parenthesis over denominator 13 end fraction

    we can summarize the sum of GP formulas as follows:

    • S = a / (1 - r) when |r| < 1
    • S = ±∞, when |r| ≥ 1
    where 'a' is the first term and 'r' is the common ratio.

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