Radius of convergence

Mathematics
Tutor: None Selected Time limit: 1 Day

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

1/(1+x^2)

May 18th, 2015

The McLaurin series (or Taylor series for x0 = 0) of the function (1 + x)α equals 

1 + αx + α(α – 1)x2/2 + … + α(α – 1)(α – 2)...(α – n + 1)xn/n! + … =  1 + ∑n=1α(α – 1)(α – 2)...(α – n + 1)xn/n!.

 Substitute α = – 1 and replace x by x2 to obtain

1/(1 + x2) = 1 – x2 + x4 – x6 + … + (–1)n – 1x2n + … = ∑n=0(–1)n – 1x2n .

Use the root test to study the convergence radius: R = 1 / lim supn→∞ |an| = 1 because |a2n| = 1 and

|a2n+1| = 0. 


May 18th, 2015

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