find-a-power-series-representation-centered-at-00-for-hx4x1x23-hx-frac4x1-x23-al

Question

Find a power series representation centered at 00 for h(x)=4x(1+x2)3 h(x) = \frac{4x}{(1 + $$x^2$)^3$$} . Also, find the interval of convergence.

Accepted Answer

∑n=0∞2(−1)n(n+2)(n+1) x2n+1\displaystyle\sum_{n=0}^{\infty}$$2(-1)^{n}(n+2)(n+1$$)\,$$x^{2n+1}$$(−1,1)\left(-1,1\right)

Answer

∑n=0∞2(−1)n(n+2)(n+1) x2n+1\displaystyle\sum_{n=0}^{\infty}$$2(-1)^{n}(n+2)(n+1$$)\,$$x^{2n+1}$$[−1,1]\left\lbrack-1,1\right\rbrack

Answer

∑n=0∞(−1)n(n+2)(n+1) x2n+1\displaystyle\sum_{n=0}^{\infty}$$(-1)^{n}(n+2)(n+1$$)\,$$x^{2n+1}$$(−1,1)\left(-1,1\right)

Answer

∑n=0∞2(−1)n(n+2)(n+1) x2n+1\displaystyle\sum_{n=0}^{\infty}$$2(-1)^{n}(n+2)(n+1$$)\,$$x^{2n+1}$$[−1,1)\left\lbrack-1,1\right)

Find a power series representation centered at 00 for h(x)=4x(1+x2)3 h(x) = \(\frac{4x}{(1 + x^2)^3}\) . Also, find the interval of convergence.