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Coefficient Sums for Infinite Polynomial Series

For polynomials of finite degree spans

P(C, k) = Σ c_{j}k^{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

To deal with the infinitude of coefficients the concept of partial sum must be introduced. The partial sum for the sequence
{c_{0}, c_{1}, …} is S_{q}=Σ_{0}sup>qc_{j} for q=0, 1, ….
This generates the sequence {S_{0}, S_{1}, …} and its convergence to a limit may be considered.

The coefficent sum for P_{1} converges to 2. The coefficent partial sums for P_{2} oscillate between 0 and 1.
For P_{3 } and P_{4} the coefficent partial sums diverge toward +∞. For P_{ } and P_{4}
the coefficent partial sums diverge toward ±∞.

The series P_{1 } 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 P_{1}, or rather, the limit of the partial sums
of the coefficients.

P_{2} 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.

P_{3 } 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.

P_{4 } and P_{5 }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.