Power Series Calculator
Instantly find the Taylor and Maclaurin power series of common functions, see the symbolic expansion, approximate any value, and get the interval of convergence. Free, fast, and built for Calculus students by Online Tools.
Power Series Expansion
Series Type
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Approx. at x
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Actual Value
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Approximation Error
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Radius of Convergence
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Interval of Convergence
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About This Tool
The Power Series Calculator finds the Taylor and Maclaurin series expansion of the most common Calculus functions, then shows you the symbolic series, a numeric approximation at any x value, the approximation error, and the interval of convergence. It is built for US Calculus II and AP Calculus BC students who need to expand functions like e^x, sin(x), cos(x), and ln(1 + x) into power series quickly and correctly, whether for homework, exam prep, or checking their own work.
A power series represents a function as an infinite sum of terms with increasing powers of x, in the form c0 + c1*x + c2*x^2 + c3*x^3 and so on. Many smooth functions can be rewritten this way, which makes them far easier to differentiate, integrate, and approximate. This tool handles the most frequently tested functions with their exact, known expansions, so the results are reliable every time.
How the Power Series Calculator Works
The calculator uses three simple inputs:
- Function f(x): pick from the most common Calculus functions, including e^x, sin(x), cos(x), ln(1 + x), 1/(1 − x), arctan(x), sinh(x), and cosh(x)
- Number of Terms: choose how many terms of the series to display (1 to 12)
- Evaluate at x: enter a value to approximate the function using the truncated series and compare it to the true value
When you click Generate Power Series, the tool builds the symbolic expansion, sums the chosen number of terms at your x value, compares that approximation to the actual function value, and reports the radius and interval of convergence.
Common Power Series Expansions
These are the standard Maclaurin series (centered at 0) that this calculator uses. They are worth memorizing for any US Calculus exam:
| Function | Maclaurin Series | Interval of Convergence |
|---|---|---|
| e^x | 1 + x + x²/2! + x³/3! + ... | (−∞, ∞) |
| sin(x) | x − x³/3! + x⁵/5! − ... | (−∞, ∞) |
| cos(x) | 1 − x²/2! + x⁴/4! − ... | (−∞, ∞) |
| ln(1 + x) | x − x²/2 + x³/3 − x⁴/4 + ... | (−1, 1] |
| 1 / (1 − x) | 1 + x + x² + x³ + ... | (−1, 1) |
| arctan(x) | x − x³/3 + x⁵/5 − ... | [−1, 1] |
| sinh(x) | x + x³/3! + x⁵/5! + ... | (−∞, ∞) |
| cosh(x) | 1 + x²/2! + x⁴/4! + ... | (−∞, ∞) |
What Is the Interval of Convergence?
A power series only equals its function for certain x values. The radius of convergence R tells you how far from the center the series stays valid, and the interval of convergence is the actual range of x values where the series converges. For example, the geometric series 1/(1 − x) only converges for x between −1 and 1, while e^x, sin(x), and cos(x) converge for every real number, giving them an infinite radius of convergence.
Benefits of Using the Power Series Calculator
- Get the exact expansion: see the symbolic Maclaurin series term by term
- Approximate any value: plug in an x and see how close the truncated series gets
- Check your error: compare the series approximation to the true value instantly
- Learn convergence: see the radius and interval of convergence for each function
- Study smarter: add more terms and watch the approximation tighten
- Completely free: no signup, runs in your browser, nothing stored
How to Use Your Results
- Increase the term count: add terms and watch the approximation error shrink toward zero
- Test convergence limits: try x values inside and outside the interval of convergence to see accuracy break down
- Verify homework: match the symbolic expansion against your hand-worked series
- Compare functions: notice how sin and cos use only odd or even powers