Power Series Calculator
Generate Taylor and Maclaurin series for common Calculus functions see the symbolic expansion term by term, approximate any value, and check the interval of convergence instantly.
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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What Is a Power Series, and Why Does It Matter in Calculus?
A power series rewrites a function as an infinite sum of terms with increasing powers of x: c₀ + c₁x + c₂x² + c₃x³ + …. It's one of the most powerful ideas in Calculus II and AP Calculus BC because it turns complicated functions into simple polynomials suddenly you can differentiate term by term, integrate term by term, and approximate function values with just a handful of additions and multiplications. This calculator generates the exact Maclaurin series (Taylor series centered at 0) for the eight most commonly tested functions, shows you the symbolic expansion, approximates the function at any x value, and reports the approximation error alongside the interval of convergence.
The distinction between a Taylor series and a Maclaurin series is just the center point. A Taylor series expands around any point a, while a Maclaurin series is the special case where a = 0. Since most Calculus textbooks and AP exams focus on Maclaurin expansions, all eight functions in this tool use center 0. If you're working with datasets where polynomial approximation is the goal, our linear regression calculator handles the simplest case a first-degree polynomial fit with slope, intercept, and R².
How Each Term Builds the Approximation
A power series converges because each additional term adds a smaller and smaller correction at least when x is inside the interval of convergence. The visual below shows what a sample expansion looks like when broken into individual term chips. Each chip represents one term with its coefficient, its power of x, and its numeric contribution at the chosen x value. Run the calculator above to populate this strip with your function's actual terms:
Notice how the contribution shrinks rapidly by Term 4, the correction is just 0.0026. That's why truncating after five terms already gives an excellent approximation for e⁰·⁵ (≈1.6487 vs. the series sum of 1.6484). The same pattern holds for sin(x), cos(x), and the other functions: the factorial denominator grows faster than any polynomial numerator, forcing each successive term toward zero. This is also why adding more terms in the calculator narrows the error and why going outside the interval of convergence breaks the pattern entirely.
Complete Maclaurin Series Reference
These are the exact expansions the calculator uses. They're worth memorizing for any Calculus exam:
| Function | Maclaurin Series | Interval of Convergence | Radius |
|---|---|---|---|
| eˣ | 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 / (1 − x) | 1 + x + x² + x³ + … | (−1, 1) | 1 |
| arctan(x) | x − x³/3 + x⁵/5 − … | [−1, 1] | 1 |
| sinh(x) | x + x³/3! + x⁵/5! + … | (−∞, ∞) | ∞ |
| cosh(x) | 1 + x²/2! + x⁴/4! + … | (−∞, ∞) | ∞ |
The interval of convergence is the range of x values where the infinite sum actually equals the function. For eˣ, sin(x), cos(x), sinh(x), and cosh(x), that range is all real numbers infinite radius. For ln(1 + x), 1/(1 − x), and arctan(x), the interval is bounded. If you evaluate the series at an x outside that interval, adding more terms makes the approximation worse, not better. The calculator reports the correct interval for each function so you know exactly where the expansion is valid. If you're also working with square roots as part of a Calculus problem, the Racine Carrée Calculator handles simplification and imaginary-number cases.
Using the Calculator for Exam Prep
- Verify your hand-worked expansions: generate the series for sin(x) or ln(1 + x) and compare the first five terms against your own derivation. If they match, you've got the pattern right.
- Understand alternating series: sin(x), cos(x), ln(1 + x), and arctan(x) alternate signs. Watch how each term flips the sign in the symbolic expansion that's the (-1)ⁿ factor at work.
- Test convergence boundaries: try x = 1.5 with 1/(1 − x). The series diverges because 1.5 is outside (−1, 1). The error will be large no matter how many terms you add a vivid demonstration of why the interval matters.
- Build intuition for error decay: increase the term count from 3 to 5 to 8 and watch the approximation error shrink. This is the core idea behind truncation error in numerical analysis.
Frequently Asked Questions
