Convergence analysis
Formula used
Partial sum check
Step-by-step solution
Convergence interval
Partial sums and function comparison
Term table
| n | Coefficient | Term | Partial sum |
|---|
Calculator inputs
Formula used
A power series usually converges around its center. The radius measures that symmetric distance. Endpoint behavior must still be checked carefully.
The ratio test works well for factorials and exponentials. The root test helps with nth powers. Numerical estimates support difficult symbolic expressions.
How to use this calculator
- Select a general, Taylor, or Maclaurin series.
- Enter the coefficient or source function.
- Set the center and starting index.
- Choose a convergence method or automatic detection.
- Enter a test value and term count.
- Press calculate to inspect every convergence step.
Example data table
| a(n) | Center | Start | Radius | Interval |
|---|---|---|---|---|
| 1/(n*4^n) | 2 | 1 | 4 | [-2, 6) |
| 1/factorial(n) | 0 | 0 | ∞ | (-∞, ∞) |
| (-1)^(n+1)/n | 0 | 1 | 1 | (-1, 1] |
| factorial(n) | 3 | 1 | 0 | {3} |
| n^2/5^n | 4 | 1 | 5 | (-1, 9) |
Common mistakes
Do not stop after finding the radius. The boundary points remain undecided. Each endpoint needs an independent series test.
Keep the series center separate from the coefficient. Check the starting index before simplifying. Small indexing errors can change endpoint behavior.
Frequently asked questions
What is a power series?
A power series is an infinite polynomial. Its terms depend on increasing powers. Convergence varies with the chosen x-value.
What is the radius of convergence?
The radius is the convergence distance from center. It may be zero or infinite. Endpoints are not decided by radius alone.
Why must endpoints be tested separately?
Ratio and root tests often become inconclusive there. One endpoint may converge while another diverges. Direct substitution reveals the correct behavior.
What means absolute convergence?
Absolute convergence survives after taking term magnitudes. It implies ordinary convergence too. Conditional convergence fails this stronger test.
Can the interval contain one point?
Yes, when the radius equals zero. Only the center may converge. Factorial coefficients often create this outcome.
When is the radius infinite?
Infinite radius means convergence for every real x. Exponential Taylor series provide a common example. Many factorial denominators create rapid decay.
Which test should I choose?
Use automatic mode for most expressions. Ratio tests suit factorials and exponentials. Root tests suit nth-power structures.
Why can numerical estimates differ?
Finite samples approximate infinite behavior. Slow convergence can mislead estimates. Increase terms and compare symbolic reasoning.
Can this generate Taylor coefficients?
Yes, supported functions are differentiated symbolically. Coefficients use derivatives at the center. Complicated functions may require simpler notation.