Asked by Dalal
Find the power series representation about the center c=0 of
(a) f(x)= 1/(1+x^2)
(b) Use part (a) to find the power series of g(x)=ln(1+x^2)
-Thank You
(a) f(x)= 1/(1+x^2)
(b) Use part (a) to find the power series of g(x)=ln(1+x^2)
-Thank You
Answers
Answered by
Steve
taking various derivatives, it is clear that
1/(1+x^2) = 1 - x^2 + x^4 - x^8 + ... for |x|<1
now, since d/dx ln(1/(1+x^2)) = 2x/(1+x^2)
2x/(1+x^2) = 2x - 2x^3 + 2x^5 - 2x^7 + ...
and
ln(1/(1+x^2)) = ∫ 2x/(1+x^2) dx = x^2 - x^4/2 + x^6/3 - x^8/4 + ...
1/(1+x^2) = 1 - x^2 + x^4 - x^8 + ... for |x|<1
now, since d/dx ln(1/(1+x^2)) = 2x/(1+x^2)
2x/(1+x^2) = 2x - 2x^3 + 2x^5 - 2x^7 + ...
and
ln(1/(1+x^2)) = ∫ 2x/(1+x^2) dx = x^2 - x^4/2 + x^6/3 - x^8/4 + ...
There are no AI answers yet. The ability to request AI answers is coming soon!
Submit Your Answer
We prioritize human answers over AI answers.
If you are human, and you can answer this question, please submit your answer.