Find the Taylor series of the following functions about x0 = 0 and its radius of convergence

1/(1+x^2)

The McLaurin series (or Taylor series for x_{0} = 0) of the function (1 + x)^{α }equals

1 + αx + α(α – 1)x^{2}/2 + … + α(α – 1)(α – 2)...(α – n + 1)x^{n}/n! + … = 1 + ∑_{n=1}^{∞}α(α – 1)(α – 2)...(α – n + 1)x^{n}/n!.

Substitute α = – 1 and replace x by x^{2} to obtain

1/(1 + x^{2}) = 1 – x^{2} + x^{4} – x^{6} + … + (–1)^{n – 1}x^{2n} + … = ∑_{n=0}^{∞}(–1)^{n – 1}x^{2n }.

Use the root test to study the convergence radius: R = 1 / lim sup_{n→∞ }|a_{n}| = 1 because |a_{2n}| = 1 and

|a_{2n+1}| = 0.

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