Section 10.6. Representations of functions as a power seriesGeometric series.∞Xn=0xn=11 − x, |x| < 1.Example 1. Find a power series representation for the function and determine the intervalof convergence.1.11 + x2.11 − x23.11 + 4x214.1 + x21 − x2Term-by-term differentiation and integration. If the power series∞Pn=0cn(x − a)nhasradius of convergence R > 0, then the function f defined byf(x) = c0+ c1(x − a) + c2(x − a)2+ ... + cn(x − a)n+ ... =∞Xn=0cn(x − a)nis differentiable (and therefore continuous) on the interval (a − R, a + R) andf0(x) = c1+ 2c2(x − a) + ... + ncn(x − a)n−1+ ... =∞Xn=1ncn(x − a)n−1Zf(x)dx = C + c0(x − a) + c1(x − a)22+ ... + cn(x − a)n+1n + 1+ ... =C +∞Xn=0cn(x − a)n+1n + 1The radii of convergence of these series are R. This does not mean that the interval of conver-gence remains the same.Example 2. Find a power series representation for the function and determine the radiusof convergence.1.1(1 + x)222. ln(1 + x)Example 3. Evaluate an indefinite integralRtan−1(x2) dx as a power series.Example 4. Use a power series to approximate the integralZ1/20tan−1(x2) dxto six decimal
View Full Document
Unlocking...