San José State University

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Thayer Watkins
Silicon Valley
USA

Coefficient Sums for
Infinite Polynomial Series

For polynomials of finite degree spans

#### P(C, k) = Σ cjkjfor j running from m up to n n and m being integers

the coefficient sum σ(P(C, 1) = Σ cj 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 Sq0sup>qcj 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 Sumsand their Limits

(To be continued.)

Now functions of