Power Series Calculator
Expand eˣ, sin x, cos x, ln(1+x), 1/(1−x), arctan x and more as power series, see each term and partial sum, and compare with the true value.
Power Series Calculator
Calculate and visualize the power series (Taylor series about any center c) of nine common functions, or of your own coefficients, with the radius and interval of convergence. A power series is an infinite sum of the form Σ aₙ(x - c)ⁿ, where aₙ are coefficients, x is the variable, and c is the center of expansion. The calculator does not accept an arbitrary formula for f(x).
Function Selection
Parameters
Power Series Calculator: Stop Expecting Magic, Start Using Partial Sums Correctly
The common first impression of a power series calculator is that you type in any function and it produces the infinite polynomial that equals that function everywhere. That is not what this tool does. This power series calculator works with nine preset functions, exponential, sine, cosine, natural log, geometric, binomial, arctangent, hyperbolic sine, and hyperbolic cosine, or with custom coefficients you supply. It cannot derive a Taylor series from an arbitrary formula you type in. What it does, it does well: it computes partial sums of the known series for those functions, lets you choose the center and the number of terms, and shows you exactly how well (or badly) that partial sum approximates the true function value at a single x.
Select a function, set the center c, choose how many terms to include. The calculator returns the partial sum, the actual function value (for the preset series), the absolute and relative errors, and a full table of every term with its contribution. There is no hidden calculation of the interval of convergence, you need to know that from the standard expansions or test it yourself. The calculator tells you the radius and interval for each preset series, but only if you check the convergence info option. For custom series, you are on your own: the ratio test or root test applied to your coefficients is the only way to find R.
- What the calculator inputs: Function selection, x value, center c (0 for Maclaurin), number of terms, decimal places, show steps toggle, show convergence toggle.
- What the calculator outputs: Partial sum S_N, actual function value, absolute error |f(x)-S_N|, relative error as percentage, individual term table with partial sums and errors, convergence info (radius and interval).
- Preset functions available: e^x, sin x, cos x, ln(1+x), 1/(1-x), (1+x)^n (binomial), arctan x, sinh x, cosh x.
- What it does NOT do: It does not accept an arbitrary f(x) formula. It does not compute the radius or interval from scratch for a custom series. It only evaluates at one x per run, no plotting or range scan.
How to Use the Power Series Expansion Calculator
Select the Function and Enter Values
Step 1: Select the function. The dropdown gives you nine common functions plus a Custom Series option. Pick one, Exponential (e^x) for example.
Step 2: Enter the x value and center c. The x value is where you want to evaluate the series. The center c is the expansion point. Setting c = 0 gives a Maclaurin series. Changing c to, say, 2 shifts the whole series: the coefficients become f^(n)(2)/n! instead of f^(n)(0)/n!. The calculator recomputes them correctly for the selected function.
Step 3: Choose the number of terms. This is the N in the partial sum S_N = Σ a_n(x-c)^n from n=0 to N-1. More terms usually mean better accuracy near the center, but not if x is outside the interval of convergence, then adding terms makes the partial sum diverge faster.
Adjust Display Options and Handle Custom Series
Step 4: Adjust display options and click Calculate Series. You can show calculation steps, convergence information, and set decimal places from 2 to 8. The steps output lists each coefficient, each term a_n(x-c)^n, and the running partial sum. The convergence info displays the radius and interval of convergence for the preset series based on the center you entered.
Custom series note: If you select Custom Series, you enter coefficients as a comma-separated list. The calculator pads with zeros if you enter fewer than the requested number of terms, and truncates if you enter more. The convergence info will state that R cannot be determined from finitely many coefficients, you must apply the ratio or root test to the general term of your series.
Reading the Term Table and Error Columns
The results table lists, for each n from 0 to N-1: the coefficient a_n, the term value a_n(x-c)^n, the partial sum S_n after including that term, and the absolute error |f(x)-S_n| (for preset series only). The error column shows how the approximation improves, or worsens, as you add terms.
For a convergent series inside its radius, the absolute error should decrease as n increases, often dramatically near the center. For example, e^x at x=1 with 10 terms gives an error below 10^-7. At x=10, the same 10 terms give an error of several hundred, the series is still convergent (radius is ∞), but you need many more terms for a good approximation that far from the center.
If you pick a function with a finite radius, like ln(1+x) at x=0.9, the error drops quickly. At x=1 (the endpoint where the alternating harmonic series gives ln 2), convergence is slow but still present: the error after 100 terms is about 1/(101) ≈ 0.01 per the alternating series estimation theorem (OpenStax Calculus Volume 2, Theorem 5.14; AP Calculus BC Unit 10.10). At x = 1.1, which is outside the interval (-1,1], the series diverges and the error column will show the partial sums growing without bound, the calculator's convergence info will flag this explicitly.
The relative error column is useful when the function value is large: an absolute error of 1 at f(x)=1000 is 0.1%, but an absolute error of 1 at f(x)=0.01 is 100%. The relative error normalises this.
What a Power Series Is (Definition, Center, Coefficients)
A power series is an infinite sum of the form Σ a_n (x-c)^n, where n runs from 0 to ∞, a_n are the coefficients, x is the variable, and c is the center of expansion (OpenStax Calculus Volume 2, Section 6.1). The series represents a function f(x) within its interval of convergence, the set of x where the infinite sum converges to a finite number.
The coefficients a_n for a Taylor series are given by f^(n)(c)/n!, the nth derivative of f evaluated at the center divided by n factorial (OpenStax Calculus Volume 2, Theorem 6.4). When c=0, this is the Maclaurin series, which is a Taylor series centered at zero (AP Calculus BC Unit 10.15). The calculator uses this definition to generate coefficients for each preset function at any center c. For example, the Maclaurin series for e^x has a_n = 1/n! because every derivative of e^x at 0 is 1. Shift the center to c=2, and a_n = e^2/n! because f^(n)(2) = e^2.
The series representation is unique: if a power series converges to a function on an interval, it must be the Taylor series of that function (OpenStax Calculus Volume 2, Section 6.3). This is why the calculator's preset functions use the known derivative formulas, they are the only possible power series expansions.
| Function | Maclaurin Series | Radius of Convergence R | Interval of Convergence |
|---|---|---|---|
| e^x | Σ x^n/n!, n=0 to ∞ | ∞ | (-∞, ∞) |
| sin x | Σ (-1)^n x^(2n+1)/(2n+1)!, n=0 to ∞ | ∞ | (-∞, ∞) |
| cos x | Σ (-1)^n x^(2n)/(2n)!, n=0 to ∞ | ∞ | (-∞, ∞) |
| ln(1+x) | Σ (-1)^(n+1) x^n/n, n=1 to ∞ | 1 | (-1, 1] |
| 1/(1-x) | Σ x^n, n=0 to ∞ | 1 | (-1, 1) |
| (1+x)^k (binomial) | Σ (k choose n) x^n, n=0 to ∞ | 1 | (-1, 1) (endpoints depend on k) |
| arctan x | Σ (-1)^n x^(2n+1)/(2n+1), n=0 to ∞ | 1 | [-1, 1] |
| sinh x | Σ x^(2n+1)/(2n+1)!, n=0 to ∞ | ∞ | (-∞, ∞) |
| cosh x | Σ x^(2n)/(2n)!, n=0 to ∞ | ∞ | (-∞, ∞) |
Where Each Preset Series Converges (And Why It Matters)
The interval of convergence is not a property of the calculator, it is a property of the series itself, determined by the ratio test or root test (OpenStax Calculus Volume 2, Sections 5.4-5.5; AP Calculus BC Units 10.8-10.9). The calculator displays the radius and interval for each preset series when you select the convergence info option, but it does not compute them on the fly. You must know that, for example, ln(1+x) converges only for -1 < x ≤ 1 (converges at x=1 by the alternating series test, diverges at x=-1 as the negative harmonic series). Choosing x=2 gives a divergent series, the calculator shows the partial sums growing and flags the value as outside the interval.
The same logic applies to the binomial series (1+x)^k. For non-integer k, the radius is 1; convergence at the endpoints depends on k. If k > -1, the series converges at x=1; if k ≥ 0, it converges at x=-1 (OpenStax Calculus Volume 2, Section 6.4). The calculator handles this correctly: for k = 1/2 (square root), the series converges at x=1 and also converges at x=-1. A common mistake is assuming the binomial series always converges for |x| < 1 without checking endpoints, the calculator's convergence info pane lists the endpoint test results.
For functions with infinite radius (e^x, sin x, cos x, sinh x, cosh x), the series converges for every real x. The partial sum approximation improves as you add terms, but the number of terms needed for a given accuracy increases as |x-c| grows. The Lagrange error bound (AP Calculus BC Unit 10.12; OpenStax Theorem 6.7) quantifies this: |R_N| ≤ M|x-c|^(N+1)/(N+1)!, where M is the maximum of the (N+1)th derivative on the interval between c and x. For e^x at c=0, this means the error bound grows exponentially with x, at x=10, even 20 terms give an error bound of about e^10/20! ≈ 0.0001, but at x=20, you need roughly 40 terms for the same bound.
What the Calculator Does Not Show (And How to Fill the Gap)
The calculator evaluates partial sums at a single x per run. It does not:
- Plot the partial sum versus the true function over a range of x.
- Compute the Lagrange error bound or the alternating series remainder bound, it only shows the actual numeric error at the chosen x.
- Derive a power series representation from a non-standard function (e.g., x^2/(1-x^3)). You must perform algebraic manipulation or substitution yourself using known series as building blocks (AP Calculus BC Unit 10.14; OpenStax Section 6.2).
To find the interval of convergence for a custom series, apply the ratio test yourself: compute L = lim |a_{n+1}/a_n| as n→∞, then R = 1/L. For example, with coefficients a_n = 1/n!, the limit is 0, so R = ∞. For a_n = 1, the limit is 1, so R = 1. The root test works similarly: L = lim sup |a_n|^(1/n). Both are described in OpenStax Calculus Volume 2, Section 6.1 and AP Calculus BC Unit 10.13.
Common Questions
What is a power series used for?
Power series are used to represent functions as infinite polynomials, which allows easier differentiation, integration, and approximation. They are the basis for Taylor and Maclaurin series, and are essential in solving differential equations, numerical analysis, and physics (OpenStax Calculus Volume 2, Chapter 6). The calculator shows you this representation in action for nine common functions.
What is the interval of convergence?
The interval of convergence is the set of all x for which the power series converges to a finite value. It is always centered at the expansion point c. The radius R is found via the ratio or root test (AP Calculus BC Unit 10.13). The calculator tells you the interval for each preset series, for example, ln(1+x) converges on (-1,1] about c=0. Outside this interval, the partial sums diverge and the series is useless.
How many terms should I use?
The number of terms needed depends on how close x is to the center c and how accurate you need the result. Near the center, 5-10 terms often give high accuracy for functions like e^x or sin x. Far from the center, or near an endpoint, you may need 50+ terms. The calculator's error columns let you experiment: add terms until the absolute error falls below your tolerance. The Lagrange error bound (AP Calculus BC Unit 10.12) gives a theoretical guarantee, but the calculator only shows the actual error at the chosen x.
What is the difference between a Taylor series and a Maclaurin series?
A Maclaurin series is a Taylor series centered at c=0 (OpenStax Calculus Volume 2, Definition 6.3). The calculator lets you set any center c. When c=0, the series is Maclaurin; when c≠0, it is the Taylor series about that point. The coefficients change accordingly: for e^x, a_n = e^c/n! for a Taylor series about c, versus a_n = 1/n! for the Maclaurin series.
Can I input my own series?
Yes. Select 'Custom Series' and enter the coefficients a_n as a comma-separated list. The calculator treats them as the coefficients of Σ a_n (x-c)^n. It cannot compute the radius of convergence from finitely many coefficients, you must apply the ratio or root test to the general term yourself. The calculator will evaluate the partial sum at your chosen x and show the individual terms, but it cannot verify that the infinite series converges there.
