Calculator
Example Data Table
| Example | Model | Inputs | Expected idea |
|---|---|---|---|
| Geometric | A∑rⁿ | A = 1, r = 0.5, n = 0, infinite | Converges to 2 |
| Centered power | A∑(x-a)ⁿ | A = 3, x = 0.25, a = 0, infinite | Converges because |x-a| < 1 |
| Exponential Taylor | A∑(x-a)ⁿ/n! | A = 1, x = 1, a = 0, infinite | Approximates e |
| Custom coefficients | ∑cₙ(x-a)ⁿ | 1, 2, 3 with x = 2, a = 0 | Evaluates 1 + 4 + 12 |
Formula Used
The general power series form is:
S = ∑ cₙ(x - a)ⁿ
Here, cₙ is the coefficient, x is the input value, and a is the center. The calculator also supports special models.
- Geometric: S = A∑rⁿ. Infinite sum is A rᵐ / (1-r), when |r| < 1.
- Centered constant coefficient: S = A∑(x-a)ⁿ. It converges when |x-a| < 1.
- Exponential Taylor: S = A∑(x-a)ⁿ / n!. The closed form is A eˣ⁻ᵃ.
- Sine Taylor: S = A∑(-1)ⁿ(x-a)²ⁿ⁺¹ / (2n+1)!.
- Cosine Taylor: S = A∑(-1)ⁿ(x-a)²ⁿ / (2n)!.
- Logarithm Taylor: S = A∑(-1)ⁿ⁺¹(x-a)ⁿ / n.
- Weighted: S = A∑(x-a)ⁿ / (n+s)ᵖ.
How to Use This Calculator
- Select the series model that matches your problem.
- Enter the coefficient, x value, center, and required index range.
- Use the ratio field for a geometric series.
- Use p and shift for weighted power series.
- Enter custom coefficients when the model is not predefined.
- Check infinite sum when you want convergence testing.
- Press calculate to show the result above the form.
- Use CSV or PDF buttons to export the computed table.
Understanding Sum of Power Series
Power Series Basics
A power series writes a function as an infinite sum. Each term uses a power of a chosen argument. That argument is often x minus a center. This calculator helps you inspect that structure before using the result in calculus, physics, or numerical work.
Why Convergence Matters
Power series can behave very differently at different x values. A small change may turn a convergent sum into a divergent one. That is why the tool reports the ratio, radius idea, endpoint behavior, and stopping reason. It also separates closed forms from numerical approximations. This makes each answer easier to audit.
Supported Series Models
The calculator supports common models. You can test a geometric series, a centered constant coefficient series, exponential terms, sine terms, cosine terms, logarithmic terms, weighted power terms, or custom coefficients. Finite sums are useful for homework, truncation studies, and polynomial approximations. Infinite sums are useful when the convergence conditions are satisfied.
Numerical Accuracy
For numerical work, the tolerance controls when a term is considered small. The maximum term count prevents endless loops. A small tolerance gives a more careful estimate. It may also require more terms. Always compare the reported term size with your accuracy needs.
Reading the Result
The result panel shows the main sum, the computed partial sum, the convergence note, and a term table. The table is helpful because it reveals sign changes, growth, decay, and cancellation. Large alternating terms can hide error. Slowly shrinking terms can require many iterations.
Closed Forms
Closed forms are shown when the selected model allows them. For example, a geometric infinite sum has a simple result when the absolute ratio is below one. Taylor models for exponential, sine, and cosine converge for every real input. The logarithm model needs stricter endpoint checks.
Custom Coefficients
Use custom coefficients when your series comes from a recurrence, data fit, or generated polynomial. Enter coefficients in order. The calculator multiplies each coefficient by the matching power of x minus the center.
Practical Review
Treat every output as a structured estimate. Check units when a model represents a physical process. Compare nearby x values to understand stability. Save the table when you need reproducible notes, classroom examples, or later review.
Final Note
This tool does not replace proof. It supports exploration. Use it to test examples, export results, and prepare clean solution steps.
FAQs
1. What is a power series?
A power series is a sum of terms using powers of x minus a center. It often represents a function near that center.
2. What does the center mean?
The center is the value a in x - a. It shifts the series around a chosen point.
3. When does a geometric series converge?
An infinite geometric series converges when the absolute value of its ratio is less than one.
4. What is the radius of convergence?
It is the distance from the center where the power series converges. Endpoint behavior may still need separate checks.
5. Why use tolerance?
Tolerance tells the calculator when a term is small enough to stop an infinite approximation.
6. Can I calculate finite sums?
Yes. Leave infinite sum unchecked. Then enter the starting index and ending index for the finite calculation.
7. How do custom coefficients work?
Enter coefficients in order. The first value is c₀, the second is c₁, and each multiplies the matching power.
8. Is the exported PDF exact?
The PDF records the displayed result and term table. It is designed for quick saving, printing, and review.