suppose that the coefficients of the power series ∑ anz^n are integers, infinitely many of wich are distinct from zero. prove that the radius of convergence is at most 1.
please steps by steps!
Set m=n+1, j=√-1, c=cos, s=sinz=r(cQ+j*sQ) => z^n=r^n[c(nQ)+j*s(nQ)]......DeMoivre's TheoremConvergent codition: |{am*r^m[c(mQ)+j*s(mQ)]}/{an*r^n[c(nQ)+j*s(nQ)}|<1上下乘[c(nQ)-j*s(nQ)],分母=an:|am*r{[c(mQ)c(nQ)+s(mQ)s(nQ)]+j[s(mQ)c(nQ)-c(mQ)s(nQ)]}/{an}|<1|am*r[c(mQ-nQ)+j*s(mQ-nQ)]/an|<1|am*r(cQ+j*sQ)/an|<1|am*z/an|<1|z|<an/am......|z|=r-an/am < r < an/am......ans
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Set m=n+1, j=√-1, c=cos, s=sinz=r(cQ+j*sQ) => z^n=r^n[c(nQ)+j*s(nQ)]......DeMoivre's TheoremConvergent codition: |{am*r^m[c(mQ)+j*s(mQ)]}/{an*r^n[c(nQ)+j*s(nQ)}|<1上下乘[c(nQ)-j*s(nQ)],分母=an:|am*r{[c(mQ)c(nQ)+s(mQ)s(nQ)]+j[s(mQ)c(nQ)-c(mQ)s(nQ)]}/{an}|<1|am*r[c(mQ-nQ)+j*s(mQ-nQ)]/an|<1|am*r(cQ+j*sQ)/an|<1|am*z/an|<1|z|<an/am......|z|=r-an/am < r < an/am......ans
這里很不錯9９9ｓhｏpｐiｎg。cｏｍ老婆很喜歡
吶僘伸咵