Is the solution correct? Question: Use the fact that 1 (1+x)2 = ∞ ∑ n=0(−1)n(n+1)xn   (converges for −1<x<1) to find a series that converges to −2 (1+x)3 (for −1<x<1). "Solution": Line 1: Notice that if f(x)= 1 (1+x)2   then f′(x)= −2 (1+x)3 . Line 2: Therefore, to find a series for −2 (1+x)3   we could differentiate the series for 1 (1+x)2 . Line 3: To find the derivative of a convergent series, we can differentiate term-by-term (i.e. differentiate one term at a time, as if we were differentiating a polynomial). Line 4: Let's expand the given series: Line 5: 1 (1+x)2 =1−2x+3x2−4x3+5x4−6x5+...  Line 6: Now differentiate both sides: Line 7: −2 (1+x)3 =−2+3⋅2x−4⋅3x2+5⋅4x3−6⋅5x4+...  Line 8: Write using sigma notation: Line 9: −2 (1+x)3 = ∞ ∑ n=0(−1)n(n+1)⋅n⋅xn  Line 10: The radius of convergence does not change when differentiating, so   ∞ ∑ n=0(−1)n(n+1)⋅n⋅xn  is a power series that converges to −2 (1+x)3 for −1<x<1.   Is the above "solution" correct? If not, in which line is the first error? First error in line 9 What is the correct final answer to the question? (Choose from the options below) Option II Option I:   ∞ ∑ n=0(−1)n(n+1)⋅n⋅xn   Option II:   ∞ ∑ n=1(−1)n(n+1)⋅n⋅xn−1    Option III:   ∞ ∑ n=0(−1)n(n+1)⋅n⋅xn+1    Option IV:   ∞ ∑ n=1(−1)n+1(n+1)⋅n⋅xn−1