Determine If The Following Sequence Converges or Diverges Calculator

Determine whether a sequence converges or diverges using our calculator. Enter your sequence terms and apply the appropriate convergence test to analyze the behavior of the sequence as n approaches infinity.

How to use this calculator

To determine if a sequence converges or diverges:

Enter the terms of your sequence in the input field. Separate terms with commas.
Select the convergence test you want to apply from the dropdown menu.
Click "Calculate" to analyze the sequence.
Review the results and interpretation.

For best results, enter at least 10 terms of your sequence. The calculator will analyze the behavior of the sequence as n approaches infinity.

Tests for convergence

Several tests can determine if a sequence converges or diverges:

Monotonic Sequence Test: A sequence is monotonic if it is either entirely non-increasing or non-decreasing.
Bounded Sequence Test: A sequence is bounded if all its terms lie within a specific range.
Limit Comparison Test: Compare the sequence to a known convergent or divergent sequence.
Ratio Test: For sequences of the form aₙ = (pₙ)/(qₙ), take the limit as n approaches infinity of |aₙ₊₁/aₙ|.
Root Test: For sequences of the form aₙ = (pₙ)/(qₙ), take the limit as n approaches infinity of the nth root of |aₙ|.

Monotonic Sequence Test: A sequence {aₙ} is monotonic if for all n ≥ N, either aₙ₊₁ ≥ aₙ or aₙ₊₁ ≤ aₙ.

Example calculations

Let's analyze the sequence: 1, 1/2, 1/3, 1/4, 1/5, ...

Enter the sequence terms: 1, 0.5, 0.333, 0.25, 0.2
Select the Monotonic Sequence Test
Click "Calculate"

The calculator will determine that the sequence is both monotonic (decreasing) and bounded (between 0 and 1), so it converges to 0.

Term (n)	Value (aₙ)	Difference (aₙ₊₁ - aₙ)
1	1.000	-0.500
2	0.500	-0.167
3	0.333	-0.083
4	0.250	-0.050
5	0.200	-0.040

Interpretation of results

Based on the test you selected, the calculator will provide one of the following results:

Converges: The sequence approaches a finite limit as n approaches infinity.
Diverges to Infinity: The sequence grows without bound as n approaches infinity.
Diverges to Negative Infinity: The sequence decreases without bound as n approaches infinity.
Oscillates: The sequence does not approach any finite limit and does not grow without bound.

If the sequence converges, the calculator will display the approximate limit value. If it diverges, it will indicate the direction of divergence.

FAQ

What is the difference between a convergent and divergent sequence?: A convergent sequence approaches a finite limit as n approaches infinity, while a divergent sequence does not approach any finite limit.
Which convergence test should I use?: The appropriate test depends on the form of your sequence. The Monotonic Sequence Test is often the simplest to apply.
Can I analyze sequences with negative terms?: Yes, the calculator can handle sequences with both positive and negative terms.
What if my sequence doesn't fit any of the standard tests?: For complex sequences, you may need to use advanced techniques like the Cauchy criterion or comparison tests.
How accurate are the results?: The calculator provides approximate results based on the terms you enter. For precise mathematical analysis, consult a calculus textbook.