Multiplying Power Series Calculator

Calculator inputs

Formula used

Let both series be centered at c.

A(x) = Σ a_i(x - c)^(i + p)

B(x) = Σ b_j(x - c)^(j + q)

A(x)B(x) = Σ d_n(x - c)^n

d_n = Σ a_i b_j, where i + p + j + q = n.

The calculator keeps terms with powers not greater than the maximum order.

How to use this calculator

1. Enter the first coefficient list in natural order.
2. Enter the second coefficient list the same way.
3. Set the expansion center used by both series.
4. Use starting powers when a series begins after zero power.
5. Choose the maximum product order to keep.
6. Enter a test variable value for a numerical check.
7. Press Submit to view the product above the form.
8. Use CSV or PDF export for records.

Example data table

Multiplying power series in physics

Why products matter

Power series appear often in physics. They describe fields, waves, potentials, motion, and small signal models. Multiplication helps combine two approximations. A transfer function may multiply an input series. A thermodynamic expansion may multiply a correction factor. A perturbation solution may need products of truncated terms.

How this tool handles terms

This calculator performs that product with the Cauchy rule. It treats each coefficient list as a series around the same center. The center can be zero or another value. Each list can also start at a chosen power. That is useful for shifted models, asymptotic work, and missing low order terms.

The result shows each product coefficient. It also gives the contributing pairs. This makes the algebra easier to audit. You can set a maximum order. Higher terms are ignored when they exceed that order. The warning explains when truncation may hide extra terms.

Numerical checking

For physics use, the numerical value is helpful. Enter a test value of the variable. The tool evaluates the two original series and the product series at that point. This check is not proof of convergence. It is a quick consistency test for a chosen range.

Power series products can change units. Each coefficient should match the power used in the model. When a series represents position, force, voltage, or energy, track dimensions carefully. The calculator keeps the arithmetic clear, but it cannot know your units.

Reporting and accuracy

The polynomial display is designed for reports. It writes terms in powers of the selected variable and center. The table is better for lab notebooks. CSV export helps move rows into spreadsheets. The PDF export gives a compact summary.

Use enough terms for the intended accuracy. If the omitted terms are large, the product may mislead. A small expansion variable usually improves accuracy. Compare successive orders when possible. Stable answers across orders suggest a safer approximation.

Many textbook examples stop after a few powers. Real calculations often need more control. This page lets you inspect every coefficient, not just the final expression. That detail helps find sign mistakes. It also helps confirm boundary condition work. Use it before substituting constants into a longer derivation.

The method is exact for entered terms. Accuracy still depends on the model and chosen truncation order in practice today.

FAQs