Multiple to Single Power Series Calculator

Formula Used

For addition: C(x) = Σ c_n(x-a)ⁿ, where c_n = Σ k_ia_i,n.

For subtraction: c_n = k₁a_1,n - Σ k_ia_i,n, for i greater than 1.

For multiplication: c_n = Σ a_1,ra_2,s..., where r + s + ... = n. This is the Cauchy product.

How to Use This Calculator

Choose how many series you want to combine. Enter coefficients from the lowest power to the highest power. Add a scale multiplier when a series is multiplied by a constant. Set the starting power when the first coefficient does not belong to the constant term.

Select addition, subtraction, or multiplication. Pick a truncation order to stop the result at a useful degree. Press the button. The single power series appears above the form, with a coefficient table and order estimate.

Understanding the Conversion

Why combine power series?

A power series often appears as one part of a longer expression. You may have one series for a function, another for a correction term, and another for a multiplied factor. Working with them separately can hide the final pattern. A single combined series gives one clean coefficient for each power. That makes comparison, approximation, and substitution easier.

Coefficient lists

This tool uses coefficient lists. The first value normally belongs to the constant term. The second belongs to the first power. The third belongs to the second power. A starting power changes that placement. For example, a list beginning at power two starts with the squared term. This is useful when a known expression has no lower order terms.

Addition and subtraction

Adding series is direct. The calculator lines up equal powers and adds their coefficients. Subtraction works the same way, but each later series is removed from the first one. Scaling happens before the final operation. This supports expressions such as two times one series minus half of another.

Multiplication

Multiplying series needs convolution. Every term in one series can multiply every useful term in another series. Only pairs or groups whose powers add to the target power are used. The calculator truncates the product at your chosen order. This keeps the output readable and avoids unnecessary higher powers.

Truncation order

The truncation order controls how many powers are shown. If you choose order eight, the result includes powers from zero through eight. The remaining terms are represented by big O notation. A higher order gives more detail. A lower order gives a faster summary for estimates and classroom checks.

Expansion center

The center defines the base term. When the center is zero, the series uses powers of x. When the center is two, the result uses powers of x minus two. All input series should share the same center for the displayed result to be valid.

Where errors happen

Most mistakes come from misaligned powers. A missing zero can shift every later coefficient. A wrong center can make the displayed base different from the original work. Product errors are also common, because several term combinations can create the same power. This calculator reduces those problems by listing every final coefficient in order. Still, the entered lists must match the source series. Review the first few terms carefully before trusting a long result. Small checks save time. They also reveal sign errors early during manual comparison.

Practical checks

Use simple values first. Test short lists before using longer lists. Watch zero coefficients, because they hold the correct power positions. For product work, choose an order high enough to capture the terms you need. Then compare the coefficient table with your handwritten steps. The table makes mistakes easier to find. It also helps you copy the final single series into homework, notes, or a larger calculation.