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Calculus/Binomial series and taylor series expansion

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Expert: Alon Mandes - 2/16/2010


Question
Hi,

How do you go about finding the Taylor series expansion of x^2/(1+x^2) using the binomial expansion for 1/(1-x)?? I know how to do the binomial expansion for 1/(1-x) but don't know where to start to find the Taylor expansion series for the 1st equation using the binomial expansion of the second.

Regards,

Answer
We know that : 1 + 1/x + 1/x^2 + 1/x^3 + 1/x^4 + .... = 1/(1-x)
we also know that : 1 - 1/x + 1/x^2 - 1/x^3 + 1/x^4 - 1/x^5 +......=1/(1+x)
Therefore :
1/(1+x^2) = 1 - 1/x^2 + 1/x^6 - 1/x^8 + 1/x^10 - 1/x^12 + ...
Note that : x^2/(x^2+1) = 1/(x^2+1) .
Thus,
The Taylor series expansion of  x^2/(1+x^2) is : 1 - 1/x^2 + 1/x^6 - 1/x^8 + 1/x^10 - 1/x^12 + ...


Alon.

Calculus

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Alon Mandes

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Kind of questions I can answer : Limits, Derivatives, Integration, Implicit functions, continuousity, differentiation ,Extremum problems, Lagrange multipliers, Gradients, Surface integrals, Multi variables functions ,Multi variables Integrals,Complex variables ,Complex functions, Curves, Trajectory integrals & Vector analyse,Divergence,Rotor & word problems. Kind of question I can't answer : Economics,Combinatorics,infinite series & convergence ,Statistics & Probabilities .

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1. I'm a team member of mathnerds (math site for answering questions) 2. I'm a team member in the Student's Union of the Technion, helping students who have problems in mathematics. 3. 2 years of experience as a math teacher in college. 4. I give free homework help for high school students in Mathematics & Physics. 5. I teach part time in collage the subjects : "Digital Signal Processing" , "Random Signals & Noise" , "Complex Functions".

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M.A in Mathematics & Bs.c in Electronics.

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