The function f(x)=ln(2−x) is represented as a power series f(x)=∑n=0∞cnxn. Find the first 4 coefficients in the power series. Find the radius of convergence R of the series.

Answer 1

Note that df/dx= 1/(2-x) = 1/2/(1-(x/2)) Note that the ∑ cxⁿ =c/(1-x) in general. So, 1/(2-x)=1/2∑(x/2)ⁿ= df/dx. Integrate to find f = 1/2∑(x/2)ⁿ⁺¹/(n+1). To find each term, calculate f for each n 0 to 3 and note the coefficients of xⁿ. Now use the ratio test to find the radius of convergence |x| < 2 .