Representing Functions as Series

11 9 representing Functions as series
Functionas Geometricseries
wehave a function wewantto representas a series canuseknownfunctions derivatives integrals of knownfunctions
geometri
x x2
ix
E.int E.Iat.Exh
and
iii
express a as a powerseries andfind theintervalofconvergence
replace in with x2
_Exa toe as Eocman don'tneedto
the points
and 1 x2 ct eat mice no


at ie E it i E i
nowmultiplyby is in
i.ie
we can findradiusofconvergence
i ii i am

we can alsointegrate differentiateon a r a r

t.es Enonexaim
Sten a
ya convergence mightchange


fix basicsigtaevignstatnfonveraenceei.lt findtheintegraloftextandnewinterval ofconvergence

the
ftp.inaii weneed to check withintegral

check 1 E.hn
ast.seviesjsnagsgeygnEmiig
ii or converges




in c IE i viii as
same
conversen




find apowerseries representation for inexes andfinditsradiusofconvergence



Iiiiiiii it iE iii iii renren.ee
en
is convergence thesameas so R