Updated Convergence & Divergence Tool

Series Convergence Calculator

Analyze infinite series using multiple numeric tests. Enter the general term an, and this calculator estimates convergence or divergence with n-th term, ratio, root, integral, p-series comparison and alternating series tests, plus partial sums.

n-th Term Test Ratio & Root Tests Integral & p-Series Alternating Series

Test Convergence & Divergence of Infinite Series

Use this Series Convergence Calculator to study infinite series of the form Σ an. The tool evaluates the general term an numerically, builds partial sums and applies standard convergence tests: n-th term test, ratio test, root test, integral-style comparison and alternating series criteria. It also estimates p-series behavior via a log-log fit.

Enter the general term an for your series. Use n as the index variable (you can also type k or i; they are treated as n). Supported functions: powers, exp, log, sqrt, sin, cos, tan, abs and combinations. The tests here are numeric and heuristic; they complement but do not replace formal proofs.

The calculator evaluates an numerically for the requested range and applies multiple tests. For alternating series, absolute-value tests check absolute convergence while the alternating test checks conditional convergence.

This tab shows numeric partial sums SN = Σ an from the chosen starting index. Use it to visualize whether the sequence of partial sums appears to settle down or grow without bound.

A series can converge absolutely (Σ |an| converges), converge conditionally (Σ an converges but Σ |an| diverges), or diverge. This tab interprets the numeric tests from the main analysis in that framework.

Absolute Convergence

Run a convergence analysis first to see whether the numeric tests suggest absolute convergence.

Conditional Convergence

For alternating or sign-changing series, the alternating test may show convergence even when absolute tests suggest divergence.

The calculator uses ratio, root and p-series tests applied to |an| to suggest absolute convergence and the alternating test on an itself to suggest conditional convergence when applicable.

Many series behave asymptotically like a p-series 1/np. This tab summarizes the log-log fit used in the main analysis to estimate p and compare your series against standard models.

Estimated p

Run a convergence analysis first to see the estimated p.

Comparison Model

The calculator fits log |an| versus log n to estimate the decay rate.

p-Series Verdict

Fit Quality Note

For ideal data, p > 1 typically indicates behavior similar to a convergent p-series; 0 < p ≤ 1 corresponds to divergent harmonic-type behavior. Real numeric data can be noisy, especially for small n, so the estimate is best used as a qualitative guide.

Series Convergence Calculator – Complete Guide to Convergence & Divergence Tests

The Series Convergence Calculator on MyTimeCalculator is designed to help you diagnose whether an infinite series Σ an converges or diverges using classic tests from calculus and real analysis. Instead of working only symbolically, the tool evaluates the general term numerically, builds partial sums and applies several standard tests side by side so that you can see how they agree or disagree.

With a single interface, you can explore series such as 1/n2, 1/n, (1/2)n, (-1)n/n, n!/2n, n/(n + 1), and many more. The calculator reports an overall numeric verdict together with detailed outcomes of the n-th term test, ratio test, root test, integral-style check, p-series comparison and alternating series criteria.

1. Infinite Series and Partial Sums

An infinite series is written as

Σn=a∞ an = aa + aa+1 + aa+2 + …

To decide whether the series converges to a finite sum, one studies the sequence of partial sums

SN = Σn=aN an.

If SN approaches a finite limit as N grows, the series converges; if SN grows without bound, oscillates wildly or has no limit, the series diverges. The calculator computes SN numerically up to a user-specified number of terms and summarizes the observed behavior in a table.

2. n-th Term Test for Divergence

The most basic test is the n-th term test: if

limn→∞ an ≠ 0,

then the series Σ an cannot converge. The converse is not true: having an → 0 is necessary but not sufficient for convergence. The harmonic series

Σn=1∞ 1/n

diverges even though 1/n → 0. The calculator estimates the behavior of an for large n numerically and reports whether the tail terms appear to approach zero or not.

3. Ratio Test

The ratio test is especially useful for series involving factorials, exponentials or powers:

L = limn→∞ |an+1 / an|.

The conclusions are:

  • If L < 1, the series converges absolutely.
  • If L > 1 or L = ∞, the series diverges.
  • If L = 1, the test is inconclusive.

The Series Convergence Calculator approximates the ratio |an+1 / an| numerically along the tail of the series and averages several of the last ratios. It then compares this estimate with 1 and reports a likely conclusion, clearly marking cases where the evidence is inconclusive.

4. Root Test

The root test considers the limit of the n-th root of the absolute value:

L = limn→∞ |an|1/n.

Again, the conclusions mirror those of the ratio test:

  • If L < 1, the series converges absolutely.
  • If L > 1, the series diverges.
  • If L = 1, the test is inconclusive.

The calculator evaluates |an|1/n for large n and averages several of the largest indices to give a numeric approximation of L. This is particularly helpful for power series and exponential-type series.

5. Integral-Style Check and p-Series Comparison

For positive, monotonically decreasing series, the integral test compares a series to an improper integral. In practice, many common series behave like a p-series

Σn=1∞ 1/np.

This converges for p > 1 and diverges for 0 < p ≤ 1. The Series Convergence Calculator performs a log-log regression on |an| versus n in the tail, estimating a decay rate p such that an behaves roughly like 1/np.

At the same time, it approximates an integral-style quantity by sampling the underlying function on a continuous grid and reports a qualitative integral test hint. Together, these diagnostics give a numeric sense of whether your series behaves like a convergent or divergent p-series.

6. Alternating Series and Conditional Convergence

Alternating series of the form

Σn=a∞ (-1)n bn,   bn ≥ 0

often converge even when the corresponding series of absolute values diverges. The classical alternating series test (Leibniz test) states that if:

  • bn is eventually decreasing, and
  • limn→∞ bn = 0,

then Σ (-1)n bn converges. The calculator looks for sign changes in an, checks whether the absolute values |an| are roughly decreasing and verifies that the tail terms appear to approach zero. It then reports whether the numeric behavior supports an alternating-series style convergence.

7. Absolute vs Conditional Convergence

A series Σ an converges absolutely if the series Σ |an| converges. It converges conditionally if Σ an converges but Σ |an| diverges. Many power series and exponential-type series converge absolutely, while alternating harmonic-like series converge only conditionally.

The Series Convergence Calculator applies ratio, root and p-series tests to |an| to assess absolute convergence, and applies the alternating series test to an itself for conditional convergence. The absolute vs conditional tab summarizes these findings in a simple narrative.

8. How to Use the Series Convergence Calculator Step by Step

  1. Identify the general term an of your series from the problem statement.
  2. Enter an in the main input box using n as the index (k or i are also accepted).
  3. Choose the starting index and the number of terms you want the calculator to analyze.
  4. Decide whether the ratio and root tests should use absolute values or signed terms.
  5. Click the analyze button to run all tests and read the overall numeric verdict.
  6. Switch to the partial sums tab to see how SN behaves term by term.
  7. Review the absolute vs conditional tab if your series is alternating or sign-changing.
  8. Use the p-series fit tab to compare your series with familiar p-series models.

9. Limitations and Good Practices

The Series Convergence Calculator is a numeric tool. It evaluates finitely many terms and uses heuristic thresholds to imitate classical convergence tests. As a result, it can occasionally be inconclusive or even misleading for cleverly constructed or very slowly convergent series.

You should always treat the numeric verdict as a guide rather than a proof. Use the calculator to build intuition, check homework results and experiment with examples, but rely on formal tests from your course or textbook for rigorous conclusions.

Related Tools from MyTimeCalculator

Series Convergence Calculator FAQs

Frequently Asked Questions

Quick answers to common questions about series convergence, divergence, numeric tests and how to interpret the results.

No. The Series Convergence Calculator is a numeric diagnostic tool. It evaluates finitely many terms and applies approximate versions of standard tests. The results are intended to guide your intuition and help you check examples, but they are not formal proofs. For rigorous work you should still use analytic tests as taught in calculus and real analysis courses.

In practice, numeric estimates can be sensitive to the number of terms, slow convergence or oscillatory behavior. If some tests hint at convergence while others are inconclusive or suggest divergence, you are likely in a borderline situation where a careful analytic test is needed. The tool reports such mixed evidence explicitly so that you know to look more closely at the theory for that series.

For many textbook series, a few dozen terms are enough to reveal the trend. Very slow convergence or divergence may require more terms, but requesting too many terms can magnify floating-point roundoff error. A practical approach is to start with 30–50 terms, then increase the count if the partial sums or test outputs look unstable or inconsistent.

If the series terms are not all positive, it is usually helpful to test absolute convergence first by applying the ratio, root and p-series tests to |an|. If Σ |an| converges, the original series converges absolutely. If absolute tests suggest divergence but the series appears to alternate with decreasing terms, the alternating test may still show conditional convergence.

Classical convergence tests have clear conditions under which they apply and many cases where they simply do not decide the question. The calculator mirrors this behavior: if the numeric estimates fall too close to the borderline values or the data are too noisy, the safest answer is “inconclusive.” In that case, you should rely on analytic techniques or try a different viewpoint on the same series.

Yes. The calculator treats standalone k and i as synonyms for n, so you can type expressions in the same style as your textbook, such as k^2/2^k or (-1)^i/i. Internally, the numeric engine maps these to n, but the overall behavior of the series is preserved.

Yes. To avoid excessive computation time and large floating-point errors, the calculator caps the analysis at a maximum number of terms. If you request more than this limit, you will see a warning asking you to reduce the range. For most practical examples in homework or teaching, the default limits are more than sufficient.

Related Calculators