1322e-20191-notes11-filled.pdf - MAT1322 C ALCULUS II E...

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MAT1322 CALCULUSIIELIZABETHMALTAIS11. Power SeriesWe have now see that some infinite seriesPanconverge.Now we considerpower serieswhich are essentially polynomials with infinitely manyterms.POWERSERIESApower series centered atx=ahas the formNotice that when we plug in a particular value, sayx=p, we getThus, a power series may be a convergent series for some values ofx.In particular, ifx=a(thecentre), then we haveIn general, the convergence of a power series will depend on thecoefficients, thecj’sIf a power series is centred ata= 0, then it looks like this:A power series defines a function!What is its domain?[email protected]p-aj2tCzfp-aPtu.owhichisjustanumericalseriesEanlikewhatwe'vealreadybeenStudying,butwithan=Cn(P-a)nforn=O,1,2,...÷Eocnca-a)"=co+Cia+adopt."=coAllpowerseriesconvergeattheircentrex=a.n§oCn(to)n=n§gCnXh=Cotgx+9×2+[3×3+...f(×)=n§jcn(×.a)nall×for
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