Question

Find the radius of convergence, R, of the series. 00 2(-1)"nx" n = 1 R= Find the interval, 1, of convergence of the series. (Enter your answer using interval notation.) I =

Answer 1

The given series is * (-1)^n x" . وني have to find the radius of convergence IR, of the the internal. I, of convergence of the series. series and we know that) The power series ² aan nao intenval of convergence of a is the interval of 2-value series that can be plugged into the pouver to gives Q convergent series the of the is R then And radius of convergence is half of the length interval of convergence. If the rodius of convergence the interval of convergence will include the interval (-R, R) radius of convergence obtained Ratio by open "The can be Test. we have to Calculate Cntl lim hed eller i L (say) And the radius of convergence L For the given series, 2(-1)^n since 2(-1)=(0)2 =0 thus , 2C-199nxn can be rewriten 2-199nxn. aus Then, ngo Lim 1 (nal cn lim 2 (217ht! ".(n+1) 24-0"n limno K-1)(1+ +) -|6 L 1:16421-1]

So L= Thus 는 . The radius of convergenes Senso is The interval of convergence is glven by (-R, R). (-1,1). Note ut check for the end points 7--1, then the series become Ź ?(-1)^.^(-1" ho Ž 2 (-1)2nn n: 5 27 [:(-1)20-1] ni This series is divergent Um 2n no does not exist. Again senio become & 21-1)" neign ni :72(-1" n This sories is dergent uimte 2(-1)"n deen not exist. so in this cousa the serien colll not converge is then, The interval of convergence for eithen endpoint. I = [(-4,1))