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Home / Doubts and Solutions/JEE/Mathematics

Expand the following in ascending powers of x 

(e^ix+e^-ix)/2

Asked by Vidushi412 30th November 2018, 1:05 PM
Answered by Expert
Answer:
begin mathsize 12px style e to the power of x space equals space 1 plus x plus fraction numerator x squared over denominator 2 factorial end fraction plus fraction numerator x cubed over denominator 3 factorial end fraction plus............ e to the power of i x end exponent space equals space 1 plus open parentheses i x close parentheses plus fraction numerator open parentheses i x close parentheses squared over denominator 2 factorial end fraction plus fraction numerator open parentheses i x close parentheses cubed over denominator 3 factorial end fraction plus fraction numerator open parentheses i x close parentheses to the power of 4 over denominator 4 factorial end fraction plus fraction numerator open parentheses i x close parentheses to the power of 5 over denominator 5 factorial end fraction plus fraction numerator open parentheses i x close parentheses to the power of 6 over denominator 6 factorial end fraction............... e to the power of i x end exponent space equals space 1 space plus space i x space minus fraction numerator x squared over denominator 2 factorial end fraction space minus space i fraction numerator x cubed over denominator 3 factorial end fraction plus fraction numerator x to the power of 4 over denominator 4 factorial end fraction space plus space i fraction numerator x to the power of 5 over denominator 5 factorial end fraction minus fraction numerator x to the power of 6 over denominator 6 factorial end fraction............... t o space infinity space space space............ left parenthesis 1 right parenthesis e to the power of negative i x end exponent space equals space 1 plus open parentheses negative i x close parentheses plus fraction numerator open parentheses negative i x close parentheses squared over denominator 2 factorial end fraction plus fraction numerator open parentheses negative i x close parentheses cubed over denominator 3 factorial end fraction plus fraction numerator open parentheses negative i x close parentheses to the power of 4 over denominator 4 factorial end fraction plus fraction numerator open parentheses negative i x close parentheses to the power of 5 over denominator 5 factorial end fraction plus fraction numerator open parentheses negative i x close parentheses to the power of 6 over denominator 6 factorial end fraction............... e to the power of negative i x end exponent space equals space 1 space minus space i x space minus fraction numerator x squared over denominator 2 factorial end fraction space plus space i fraction numerator x cubed over denominator 3 factorial end fraction plus fraction numerator x to the power of 4 over denominator 4 factorial end fraction space minus space i fraction numerator x to the power of 5 over denominator 5 factorial end fraction minus fraction numerator x to the power of 6 over denominator 6 factorial end fraction............... t o space infinity space space space...... space...... left parenthesis 2 right parenthesis F r o m space left parenthesis 1 right parenthesis space a n d space left parenthesis 2 right parenthesis comma space w e space w r i t e space space space fraction numerator e to the power of i x end exponent plus e to the power of negative i x end exponent over denominator 2 end fraction space equals space 1 space minus space fraction numerator x squared over denominator 2 factorial end fraction space plus space fraction numerator x to the power of 4 over denominator 4 factorial end fraction space minus fraction numerator x to the power of 6 over denominator 6 factorial end fraction plus.................. w e space h a v e space u s e d space i squared space equals space minus 1 space comma space space i cubed space equals space minus i comma space i to the power of 4 space end exponent equals space 1 comma space i to the power of 5 space equals space i space comma space i to the power of 6 equals space minus 1 space space space e t c space space i n space t h e space a b o v e space s e r i e s space e x p a n s i o n end style
Answered by Expert 1st December 2018, 8:31 AM
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