X Suppose f(x ) = 2x + 1 ( a) use the fact that the geometric series 1-x = to find the power representation for f(x ) centered at 0. (b) state the radius of convergence of the power series.In a geometric series, each term in the sum is obtained from the previous term by multiplying by r. Cark-1 = at artart ... tap-1+... k = 1 Let's figure out for which values of r the geometric series converges. Recall that Sn = a1 + a2 + a3 + ... + an is the nth partial sum. If r = 1, we have sn = a ta tata+... ta=na. Since lim sn = lim na = oo, then the geometric series n -+ 00 Cark-1 diverges for r = 1. n-too Now if r # 1, we have Sn = at art art art ... + arn- 1 Multiplying both sides by r and shifting terms one space to the right: rSn = ar tartars + ... + arn - 1 + arn Subtracting the second equation from the first we obtain (1 -r)sn = a - arn Solving for Sn gives sn = a(1 - rn) 1 -r Now, if r > 1, then lim r" = co and if r ark-1 diverges for |r| 2 1. K = 1 If r