Calculator Inputs
Formula Used
The calculator studies a power series in the form Σ an(x − c)n.
For the ratio test, use L = lim |an+1 / an|. The radius is R = 1 / L. If L is zero, R is infinite. If the terms grow too fast, R can be zero.
The interval starts with |x − c| < R. Then test x = c − R and x = c + R separately.
For Σ ((x − c) / S)n / np, both endpoints converge absolutely when p > 1. The left endpoint is conditional when 0 < p ≤ 1. The right endpoint diverges in that same range.
How to Use This Calculator
- Select the coefficient pattern that matches your series.
- Enter the center c and scale S. The scale often controls the radius.
- Enter p when using n-power templates.
- Use the custom ratio option when you already know L.
- Add a test x value to check a specific point.
- Press calculate. Review the radius, interval, endpoints, table, and graph.
- Use the CSV or PDF button to export your work.
Example Data Table
| Series model | Center c | Scale S | p | Radius | Interval result |
|---|---|---|---|---|---|
| Σ ((x − 2) / 3)n / n2 | 2 | 3 | 2 | 3 | [-1, 5] |
| Σ ((x + 1) / 4)n / n | -1 | 4 | 1 | 4 | [-5, 3) |
| Σ n-2((x − 1) / 2)n | 1 | 2 | -2 | 2 | [-1, 3] |
| Σ (x − c)n / n! | 0 | N/A | N/A | ∞ | (-∞, ∞) |
Power Series Learning Guide
Power Series Meaning
A power series is an infinite polynomial around a center. It has coefficients, powers, and a center value. The basic form is sum a_n times x minus c to the power n. The center c fixes the point where the series is built. The coefficients decide how quickly terms shrink or grow.
Radius of Convergence
The radius tells where the series converges absolutely. Inside the radius, every allowed x gives a stable sum. Outside the radius, terms usually grow or fail to approach zero. The ratio test is the common method. If L is the limit of the absolute coefficient ratio, then R equals one over L. If L is zero, the radius is infinite. If L is infinite, the radius is zero.
Interval of Convergence
The interval uses the radius and center. It starts at c minus R and ends at c plus R. Interior points converge for a normal power series. Endpoints need separate tests. They may converge absolutely, converge conditionally, or diverge. This calculator checks standard endpoint patterns. It also marks custom endpoints when a direct endpoint test is needed.
Endpoint Testing
Endpoint testing often becomes a p-series or alternating p-series. A positive p-series converges when p is greater than one. It diverges when p is less than or equal to one. Alternating terms can converge conditionally when terms decrease to zero. That is why the left and right endpoints can behave differently.
Practical Use
Use the calculator to compare templates before solving by hand. Enter the center, scale, exponent, amplitude, and test value. The result shows radius, interval, endpoint status, and test point status. The table displays terms and partial sums. The chart shows the ratio boundary. Export options help you save work for reports, classes, and notes.
Study Benefits
Power series appear in calculus, differential equations, physics, and approximation. They turn difficult functions into manageable polynomial expressions. Knowing the interval is essential. A series may work perfectly near its center, yet fail at distant values. A careful interval check prevents wrong approximations and supports better mathematical decisions. Good endpoint habits make advanced calculus problems much easier. They also reduce careless test mistakes.
FAQs
1. What is a power series?
A power series is an infinite sum of terms built from powers of x minus a center value. It behaves like an infinite polynomial near that center.
2. What is the radius of convergence?
The radius is the distance from the center where the series converges absolutely. Inside that distance, the series is stable. Outside it, the series diverges.
3. Why do endpoints need separate tests?
The ratio test usually gives no decision at endpoints. Each endpoint may become a p-series, alternating series, or another special case.
4. What does conditional convergence mean?
Conditional convergence means the original series converges, but the absolute value series diverges. Alternating endpoint series often have this behavior.
5. Can the radius be infinite?
Yes. Series with factorial denominators often converge for every real x. In that case, the interval is all real numbers.
6. Can the radius be zero?
Yes. Series with factorial growth in the numerator may converge only at the center. The interval then contains only that center value.
7. What is the role of the center c?
The center shifts the interval. If the radius is R, the first interval boundary is c minus R, and the second is c plus R.
8. What should I do for a custom series?
Find the ratio limit L from the coefficients. Enter L in the custom option. Then test both endpoints separately by hand if needed.