A formal power series over R is a formal infinite sum f = X[infinity] n=0 anxn, where the coefficients an ∈ R. We add power series term-by-term, and two power series are the same if all their coefficients are the same. (We don’t plug numbers in for x, because we don’t want to worry about issues with convergence of the sum.) There is a vector space V whose elements are the formal power series over R. There is a derivative operator D ∈ L(V ) defined by taking the derivative term-by-term: D X[infinity] n=0 anxn ! = X[infinity] n=0 (n + 1)an+1xn What are the eigenvalues of D? For each eigenvalue λ, give a basis of the eigenspace E(D, λ). (Hint: construct eigenvectors by solving the equation Df = λf term-by-term.)