Calculator
Formula Used
The alternating series test studies a series written as Σ(-1)^n b_n or Σ(-1)^(n+1)b_n, where b_n is positive.
If b_(n+1) ≤ b_n after the start index and lim b_n = 0, then the alternating series converges.
After N used terms, the error is bounded by the next positive term: |R_N| ≤ b_(N+1).
The calculator also estimates a partial sum: S_N = a_1 + a_2 + ... + a_N, where each a_n includes its alternating sign.
How to Use This Calculator
- Enter the positive magnitude b_n. Do not enter the alternating sign.
- Select whether the first listed term is positive or negative.
- Set the starting index, sample size, and partial sum terms.
- Adjust the tolerance values when the sequence changes slowly.
- Press Calculate. The result appears below the header and above the form.
- Use the CSV or PDF buttons to save your report.
Example Data Table
| Example | b_n expression | Start | Expected note |
|---|---|---|---|
| Alternating harmonic | 1/n | 1 | Converges conditionally by the test. |
| Alternating geometric | pow(0.5,n) | 1 | Terms decrease quickly toward zero. |
| Nonzero limit case | n/(n+1) | 1 | The zero limit condition fails. |
| Slow zero case | 1/sqrt(n) | 1 | Increase the limit index for better estimates. |
Understanding the Alternating Series Test
An alternating series changes sign from term to term. A common form is (-1)^n b_n or (-1)^(n+1) b_n. The test studies the positive size b_n. It does not study the signed term alone.
What the Test Checks
The alternating series test needs two main facts. First, b_n should not increase after the chosen starting point. Second, b_n should approach zero as n grows. When both facts hold, the alternating series converges. The calculator checks these ideas with a numerical sample.
Why Monotonic Terms Matter
Decreasing positive terms make each new correction smaller. The partial sums move above and below the final value. The movement shrinks as the terms shrink. This creates a stable trap around the true sum. That is why the next omitted term becomes an error bound.
Using the Remainder Bound
If the test passes, the error after N terms is at most b_(N+1). This is useful for planning. You can choose a target error. Then the tool searches for the first term small enough. That term count estimates how many terms you need.
Limits and Numerical Care
A calculator cannot prove every symbolic limit. It estimates behavior from large index values. Slow sequences may need a larger test index. A sequence like 1 over square root n approaches zero slowly. Increase the test index when the result looks uncertain.
Conditional and Absolute Behavior
The alternating series test proves convergence for signed alternating sums. It does not prove absolute convergence. For absolute convergence, the positive series sum b_n must also converge. The harmonic alternating series converges conditionally, because the positive harmonic series diverges.
Practical Study Uses
This calculator helps students compare examples. It also helps teachers create quick tables. You can export results for notes. You can check partial sums, next-term bounds, and sample term behavior in one place. Always support numerical results with class rules or symbolic work when a formal proof is required.
Reading the Output
The result card separates pass checks from warnings. It lists the sign pattern, first terms, partial sum, and estimated limit. A pass means the sampled conditions match the test. A warning means you should change options or verify the sequence by hand before final use.
FAQs
What is an alternating series?
It is a series whose terms change sign in order. The signs may follow positive, negative, positive, negative, or the reverse pattern.
What should I enter as b_n?
Enter only the positive magnitude of the term. For the alternating harmonic series, enter 1/n instead of (-1)^(n+1)/n.
Does this calculator prove convergence?
It gives a numerical test report. Formal proof may still need algebraic limit work and a written monotonicity argument.
Why did the limit test fail for a slow sequence?
The large index may not be large enough. Increase the limit test index or relax the tolerance when the term approaches zero slowly.
What is the remainder bound?
When the test passes, the error after N terms is at most the next positive term. This is the Leibniz error estimate.
Can I test absolute convergence here?
The calculator shows an absolute partial sum. It does not prove absolute convergence. Use comparison, ratio, root, or integral tests separately.
Which functions are supported?
You can use sqrt, log, exp, sin, cos, tan, abs, pow, and basic operators. Use * for multiplication.
Why use CSV and PDF exports?
CSV is useful for spreadsheets. PDF is useful for sharing a quick report with the result, settings, formula, and sample terms.