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Term-by-term operations for power series

Within the common disk of convergence, power series can be added, scaled, and multiplied by operating on coefficients.

Term-by-term operations for power series: Let

∑n=0∞an(x−x0)nand∑n=0∞bn(x−x0)n\sum_{n=0}^\infty a_n (x-x_0)^n \quad\text{and}\quad \sum_{n=0}^\infty b_n (x-x_0)^n

be with radii of convergence RaR_a and RbR_b. Set R=min⁡(Ra,Rb)R=\min(R_a,R_b). Then for every ∣x−x0∣<R|x-x_0|<R:

  • (Addition and scalar multiplication) The series ∑n=0∞(an+bn)(x−x0)n\sum_{n=0}^\infty (a_n+b_n)(x-x_0)^n and ∑n=0∞(λan)(x−x0)n\sum_{n=0}^\infty (\lambda a_n)(x-x_0)^n converge, and
    ∑n=0∞(an+bn)(x−x0)n=∑n=0∞an(x−x0)n+∑n=0∞bn(x−x0)n.\sum_{n=0}^\infty (a_n+b_n)(x-x_0)^n = \sum_{n=0}^\infty a_n(x-x_0)^n + \sum_{n=0}^\infty b_n(x-x_0)^n.
  • (Multiplication) If cn=∑k=0nakbn−kc_n=\sum_{k=0}^n a_k b_{n-k} (the coefficient ), then ∑n=0∞cn(x−x0)n\sum_{n=0}^\infty c_n (x-x_0)^n converges and equals the product of the two sums:
    (∑n=0∞an(x−x0)n)(∑n=0∞bn(x−x0)n)=∑n=0∞cn(x−x0)n.\left(\sum_{n=0}^\infty a_n(x-x_0)^n\right)\left(\sum_{n=0}^\infty b_n(x-x_0)^n\right) = \sum_{n=0}^\infty c_n(x-x_0)^n.
Remarks

These operations justify treating power series like “infinite polynomials” on their common disk of convergence and connect directly to results on and .