Sums Encoded by Product Coefficients

From Products to Sums

A power series can encode a sequence of coefficients. Multiplying two such series combines their coefficients: the coefficient of a given power is the sum of all products whose powers add to that power. This convolution can reveal a useful formula for a sequence, or turn a series sum into the value of a known function.

The key is to keep two tasks distinct. First, determine the coefficient created by the product. Then, evaluate the resulting identity only at arguments where the series and product are valid. The examples below use logarithmic and arctangent series to demonstrate both steps.

Why the Product Coefficients Work

For polynomials, the coefficient rule follows by distributing each term of the first factor across every term of the second. For example, a term \(a_jx^j\) multiplied by \(b_kx^k\) contributes \(a_jb_kx^{j+k}\). To form the coefficient of \(x^n\), the indices must satisfy \(j+k=n\), giving the finite sum in the definition.

For power series, the same coefficient rule applies within a disk where both series converge absolutely. Absolute convergence allows the terms in the product to be grouped according to their total power without changing the sum.

To see why regrouping is justified, consider the sum of the absolute values of all pairwise product terms. It factors as \[ \sum_{j=0}^{\infty}\sum_{k=0}^{\infty}|a_jb_kx^{j+k}| = \left(\sum_{j=0}^{\infty}|a_jx^j|\right) \left(\sum_{k=0}^{\infty}|b_kx^k|\right), \] which is finite under the stated assumption. Thus the double sum can be regrouped by \(n=j+k\); for each \(n\), the contributing pairs are exactly \(j=0,\ldots,n\) and \(k=n-j\). This gives the theorem’s coefficient formula and product identity.

A Logarithm Times a Geometric Series

For \(|x|<1\), the logarithm series and geometric series are \[ -\ln(1-x)=\sum_{k=1}^{\infty}\frac{x^k}{k}, \qquad \frac{1}{1-x}=\sum_{j=0}^{\infty}x^j. \] Both series converge absolutely in this interval, so their product can be expanded using convolution coefficients.

Squaring the Logarithm Series

A product of two copies of the logarithm series gives a different coefficient pattern. For \(n\geq 2\), the coefficient of \(x^n\) in \(\bigl(-\ln(1-x)\bigr)^2\) is \[ \sum_{k=1}^{n-1}\frac{1}{k(n-k)}. \] The index starts at \(1\) because the logarithm series has no constant term.

This finite sum can be simplified. The identity \[ \frac{1}{k(n-k)}=\frac{1}{n}\left(\frac{1}{k}+\frac{1}{n-k}\right) \] holds because the numerator on the right, after combining fractions, is \((n-k)+k=n\). Summing over \(k=1,\ldots,n-1\), the first reciprocal sum is \(H_{n-1}\); reversing the order in the second gives another \(H_{n-1}\). Hence the coefficient equals \(2H_{n-1}/n\).

Products of Arctangent Series

For \(|x|<1\), the arctangent series is \[ \arctan x=\sum_{k=0}^{\infty}\frac{(-1)^k x^{2k+1}}{2k+1}. \] At such \(x\), it converges absolutely. Squaring the series therefore gives a valid convolution. The product of the powers \(x^{2k+1}\) and \(x^{2m+1}\) is \(x^{2(k+m+1)}\), so terms contributing to \(x^{2n+2}\) have \(k+m=n\).

Common Pitfalls

A frequent error is to multiply corresponding coefficients instead of collecting every pair whose powers add to the target power. For example, the coefficient of \(x^3\) in \(\bigl(-\ln(1-x)\bigr)^2\) is not \(1/3^2\). There is no \(x^3/3\) term paired with a constant term, because the logarithm series has zero constant term. The actual contributing pairs have powers \(1\) and \(2\), or \(2\) and \(1\): \[ \frac{1}{1\cdot2}+\frac{1}{2\cdot1}=1. \]

Another mistake is to substitute an endpoint without checking convergence. For example, the product identity involving \(\sum H_nx^n\) was established for \(|x|<1\). It does not license setting \(x=1\). Check the convergence domain of the series before evaluating its product, and make sure the chosen argument lies within that domain.

Why Product Coefficients Are Useful

Convolution coefficients often have a recognizable structure even when the original product does not look like a sum. A coefficient calculation can express a complicated finite sum in terms of harmonic numbers, or describe an entire sequence through a compact formula. Once the product is identified with a known function, evaluating inside the convergence interval can then sum an infinite series of those coefficients.

The method is systematic: write the known series, identify which pairs of powers contribute, simplify the finite coefficient sum, and only then choose an admissible value of the variable. Keeping these steps separate helps prevent both coefficient errors and unjustified endpoint substitutions.