For polynomials of finite degree spans

P(C, k) = Σ c_jk^j
for j running
from m up to n
n and m being integers

the coefficient sum σ(P(C, 1) = Σ c_j has interesting properties. For polynomials of infinite degree spans the coefficient sum may or may not be defined. For examples, consider

P₁ = 1 + (1/2)k + (1/4)k² + (1/8)k³ + …
P₂ = 1 − k + k² − k³ + …
P₃ = 1 + k + k² + k³ + …
P₄ = 1 + 2k + 4k² + 8k³ + …
P₅ = 1 − 2k + 4k² − 8k³ + …

To deal with the infinitude of coefficients the concept of partial sum must be introduced. The partial sum for the sequence {c₀, c₁, …} is S_q=Σ₀sup>qc_j for q=0, 1, …. This generates the sequence {S₀, S₁, …} and its convergence to a limit may be considered.

The coefficent sum for P₁ converges to 2. The coefficent partial sums for P₂ oscillate between 0 and 1. For P₃ and P₄ the coefficent partial sums diverge toward +∞. For P and P₄ the coefficent partial sums diverge toward ±∞.

The series P₁ is equivalent to 1/(1−k/2). At k=1 this function is equal to 2, which is the same as the coefficent sum for P₁, or rather, the limit of the partial sums of the coefficients.

P₂ is equivalent to the function 1/(1+k), which at k=1 has the value (1/2). This intrigingly is equal to the average of the 0 and 1 the partial sums oscillate between.

P₃ is equivalent to the function 1/(1−k), which diverges toward +∞ as k→1. This corresponds to what happens to the partial sums of the coefficients of the series.

P₄ and P₅ are equivalent to the functions 1/(1−2k) and 1/(1+2k), respectively. At k=1 these functions have the values −1 and (1/3), respectively, which in no way correspond to what happens to the partial sums of the coefficients of the series.

Functions of the Partial Sums
and their Limits

(To be continued.)

Now functions of