Question Number 55634 by gunawan last updated on 01/Mar/19
$$\mathrm{If}\:\underset{{x}\rightarrow{c}} {\mathrm{lim}}\:\frac{{a}_{\mathrm{0}} +{a}_{\mathrm{1}} \left({x}−{c}\right)+{a}_{\mathrm{2}} \left({x}−{c}\right)^{\mathrm{2}} +…+{a}_{{n}} \left({x}−{c}\right)^{{n}} }{\left({x}−{c}\right)^{{n}} }=\mathrm{0} \\ $$$$\mathrm{then}\:{a}_{\mathrm{0}} +{a}_{\mathrm{1}} +{a}_{\mathrm{2}} +..+{a}_{{n}} =.. \\ $$
Answered by tanmay.chaudhury50@gmail.com last updated on 01/Mar/19
$${t}={x}−{c} \\ $$$$\underset{{t}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{{a}_{\mathrm{0}} +{a}_{\mathrm{1}} {t}+{a}_{\mathrm{2}} {t}^{\mathrm{2}} +{a}_{\mathrm{3}} {t}^{\mathrm{3}} +…+{a}_{{n}} {t}^{{n}} }{{t}^{{n}} } \\ $$$$\:{for}\left(\frac{\mathrm{0}}{\mathrm{0}}\right)\:{form}\:{a}_{\mathrm{0}} =\mathrm{0} \\ $$$$\underset{{t}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{{a}_{\mathrm{1}} +\mathrm{2}{a}_{\mathrm{2}} {t}+\mathrm{3}{a}_{\mathrm{3}} {t}^{\mathrm{2}} +…+{na}_{{n}} {t}^{{n}−\mathrm{1}} }{{nt}^{{n}−\mathrm{1}} } \\ $$$${to}\:{make}\:\left(\frac{\mathrm{0}}{\mathrm{0}}\right){form}\:{a}_{\mathrm{1}} \rightarrow\mathrm{0} \\ $$$${so}\:{in}\:{my}\:{view}\:\underset{{i}=\mathrm{0}} {\overset{{n}} {\sum}}{a}_{{i}} =\mathrm{0}\: \\ $$$${let}\:{others}\:{check}… \\ $$
Commented by gunawan last updated on 01/Mar/19
$$\mathrm{yes}\: \\ $$$$\mathrm{thank}\:\mathrm{you}\:\mathrm{Sir} \\ $$
Commented by tanmay.chaudhury50@gmail.com last updated on 01/Mar/19
$${most}\:{welcome}…{pls}\:{check}\:{the}\:{others}\:{solved}\:{problem}\:{or} \\ $$$${upload}\:{the}\:{answer}\:{only}…{not}\:{details}… \\ $$