STEM Calculators

Power Series Differentiator or Integrator

Math Calculus • Infinite Series and Sequences

Written by STEM Calculators Team Published Updated
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4. Power Series Differentiator Or Integrator
Differentiate or integrate a power series term-by-term (within its radius of convergence). Shows the transformed series, radius/interval info, and a zoomable “radius circle” plot.
Inputs
Optional helper: tries to find a smallest \(n\) (up to a cap) so the next-term magnitude at \(x_0\) is \(\le\varepsilon\).
Series definition
The tool assumes the series is \(\sum_{n=n_0}^{\infty} a_n (x-a)^n\). Enter \(a_n\) using variable n. Constants: pi, e. Supported: + − * / ^, parentheses, sin cos tan, sqrt abs exp ln/log.
For a power series, term-by-term differentiation/integration keeps the same radius \(R\) (endpoints may differ).
Use a number (e.g., 1, 2.5). You may also enter inf.
Used only for \(\int\). Leave as 0 to omit.
If provided, shows a numeric “next-term size” at \(x_0\).
Used only for numeric diagnostics (not for symbolic series output).
Click a preset to load and solve.
Ready
Graph
Drag to pan • wheel/pinch to zoom. Shows the radius circle centered at \(a\) on the real axis.
x: 0, y: 0
Results & steps
Click Calculate.

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Frequently Asked Questions

How do you differentiate a power series term-by-term?

For sum a_n (x-a)^n, the term-by-term derivative is sum n a_n (x-a)^(n-1), using the same center a. This transformation is valid for |x-a| < R, where R is the radius of convergence.

How do you integrate a power series term-by-term?

For sum a_n (x-a)^n, the term-by-term integral is sum a_n (x-a)^(n+1)/(n+1) + C. It is valid for |x-a| < R, and C is the integration constant.

Does differentiation or integration change the radius of convergence?

For a power series, term-by-term differentiation and integration keep the same radius R. Endpoint behavior at x=a±R can differ and must be checked separately.

What does the target epsilon option measure at x0?

It uses a numeric next-term magnitude computed at x0 and searches for a smallest n (up to a cap) such that the next-term size is less than or equal to epsilon. This is a practical helper for choosing how many terms are needed for a desired tolerance at that point.

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