Conditional Convergence Calculator

Calculator Inputs

Coefficient c

Power exponent p

Log exponent q

Power shift s

Log shift t

Starting index N

Phase

Displayed sample terms

Graph points

Decimal precision

This tool studies the alternating family Σ (-1)^n+phase c / ((n+s)^p(ln(n+t))^q). It applies standard p-series, logarithmic comparison, and alternating-series logic.

Formula Used

The calculator analyzes the infinite series a_n = (-1)^n+phase c / ((n+s)^p(ln(n+t))^q), using the natural logarithm.

It tests absolute convergence by studying Σ |a_n|. For this family, the absolute series converges when p > 1, or when p = 1 and q > 1.

It tests conditional convergence when the alternating version converges by the alternating-series test, but the absolute series diverges. That typically occurs here when 0 < p < 1, or when p = 1 and q ≤ 1.

If terms do not approach zero, the series is marked divergent because no infinite series can converge without that limit condition.

How to Use This Calculator

Enter the coefficient, power exponent, and logarithmic exponent.
Choose shifts for the power and logarithm parts.
Set the starting index and alternating phase.
Select how many sample terms and graph points to display.
Click Analyze Series to see the classification above the form.
Review the graph, term table, and remainder estimate.
Use the export buttons to save a CSV or PDF copy.

Example Data Table

Example Series	c	p	q	t	Phase	Expected Result
Σ (-1)ⁿ⁺¹ / n	1	1	0	1	1	Conditionally Convergent
Σ (-1)ⁿ⁺¹ / n²	1	2	0	1	1	Absolutely Convergent
Σ (-1)ⁿ⁺¹ / √n	1	0.5	0	1	1	Conditionally Convergent
Σ (-1)ⁿ⁺¹	1	0	0	1	1	Divergent
Σ (-1)ⁿ⁺¹ / (n ln(n+1))	1	1	1	1	1	Conditionally Convergent

Frequently Asked Questions

1. What does conditional convergence mean?

A series is conditionally convergent when the alternating series converges, but the series formed from absolute values diverges. The sign pattern is essential for convergence.

2. What does absolute convergence mean?

Absolute convergence means Σ|a_n| converges. This is stronger than conditional convergence, and it guarantees the original series also converges.

3. Why must the terms approach zero?

If a_n does not approach zero, partial sums cannot settle to a finite limit. This is a necessary test for every infinite series.

4. Which logarithm does the calculator use?

It uses the natural logarithm, written ln. That is the standard form for convergence rules involving logarithmic corrections.

5. Can this calculator analyze any symbolic series?

No. It is designed for a parameterized alternating-series family with power and logarithmic factors. It does not perform general symbolic algebra on arbitrary formulas.

6. Why do shifts matter?

Shifts change the starting behavior and keep expressions defined. For example, the logarithmic term requires n + t > 1 whenever q is positive.

7. What is the remainder bound shown in results?

For a convergent alternating series, the truncation error is bounded by the next omitted term’s magnitude. The calculator reports that estimate when appropriate.

8. Why show partial sums on a graph?

The graph makes convergence easier to see. Stable oscillation with shrinking jumps usually indicates convergence, while wandering or persistent jumps suggest divergence.