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

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MAT1322 CALCULUSIIELIZABETHMALTAIS11. Power Seriesâ‡§We have now see that some infinite seriesPanconverge.â‡§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 haveâ€¢In general, the convergence of a power series will depend on thecoefficients, thecjâ€™sâ€¢If a power series is centred ata= 0, then it looks like this:â€¢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,...Ã·Eocnca-a)"=co+Cia+adopt."=coâ‡’Allpowerseriesconvergeattheircentrex=a.nÂ§oCn(to)n=nÂ§gCnXh=Cotgx+9Ã—2+[3Ã—3+...f(Ã—)=nÂ§jcn(Ã—.a)nâ†³allÃ—for
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