If the Taylor Series of ln ⁡ ( x ) is known: ln ⁡ ( x ) = ( x − 1 ) − 1 2 ( x − 1 ) 2 + 1 3 ( x − 1 ) 3 − 1 4 ( x − 1 ) 4 + 1 5 ( x − 1 ) 5 − . . .Can one find the Taylor series of f ( x ) = x 1 − x 2 by manipulating the Taylor series of ln ⁡ ( x )?

Question

If the Taylor Series of   ln  ⁡  (  x  ) is known:  ln  ⁡  (  x  )  =  (  x  −  1  )  −      1    2    (  x  −  1      )    2    +      1    3    (  x  −  1      )    3    −      1    4    (  x  −  1      )    4    +      1    5    (  x  −  1      )    5    −  .  .  .Can one find the Taylor series of  f  (  x  )  =      x          1      −              x        2            by manipulating the Taylor series of   ln  ⁡  (  x  )?

artirw9f · Accepted Answer

As an alternate without using   ln  ⁡  x. Make use of the geometric series:      1          1      −      x        =      ∑          n      =      0        ∞        x    n      ⟹        1          1      −              x        2              =      ∑          n      =      0        ∞        x          2      n          ⟹        x          1      −              x        2              =      ∑          n      =      0        ∞        x          2      n      +      1

kunguwaat81 · Accepted Answer

Hint:  ∫            x              1        −                  x          2                      d    x    =  −      1    2    ln  ⁡      (    1    −          x      2        )

If the Taylor Series of ln(x) is known: ln(x)=(x−1)−1/2(x−1)^2+1/3(x−1)^3−1/4(x−1)^4+1/5(x−1)^5−...Can one find the Taylor series of f(x)=x/1−x^2 by manipulating the Taylor series of ln(x)?

Answered question

Answer & Explanation

New Questions in Calculus 2