Infinite Series Calculator with Steps Online Free

Add a half, then a quarter, then an eighth, then a sixteenth, and keep going. After ten terms you have 0.9990234375, after twenty terms you are within a millionth of 1, and if you could add forever you would reach exactly 1, never passing it. That is an infinite series: a sum with infinitely many terms that nevertheless settles on a finite total. It is one of the strangest and most useful ideas in mathematics, because it turns an endless process into a number you can use, and it underlies everything from compound interest calculations to the way computers evaluate trigonometric functions. The infinite series calculator at the top of this page adds terms one by one, shows every partial sum as a real step, and gives a cautious indication of whether the sums appear to be settling, without ever pretending to know a closed-form sum it has not computed.

The key idea is the partial sum. You cannot add infinitely many terms in practice, so you add the first N terms and call that the Nth partial sum, written S_N. If the partial sums approach a fixed value as N grows, the series converges to that value; if they grow without bound, or oscillate without settling, the series diverges. The calculator shows the partial sums explicitly, term by term, so you can watch convergence happen, or watch it fail to happen, instead of taking a textbook’s word for it. That watching is the pedagogical point of the tool: convergence is a behaviour of a sequence of numbers, and seeing the sequence is more instructive than being told the answer.

What It Is

An infinite series is written as the sum, from n = 1 to infinity, of a term formula a(n), for example 1/n², which gives 1 + 1/4 + 1/9 + 1/16 and so on. The term formula is the rule that generates each term from its position n, and the calculator accepts any term formula you can type using n, numbers, operators, brackets, and the usual functions. The partial sum after N terms is the sum of the first N terms, and the calculator computes it exactly, by adding real numbers, for the N you choose, up to one thousand terms. It also shows each term and each running total when N is one hundred or fewer, so the steps are visible, and it shows the first twenty terms when N is larger, with the final total still computed over all N terms.

Convergence and divergence are the two possible fates of a series, and they are worth distinguishing carefully, because intuition is unreliable here. A series whose terms shrink to zero can still diverge, as the harmonic series 1 + 1/2 + 1/3 + 1/4 shows: its terms tend to zero, but its partial sums grow without bound, slowly, passing 10 after about twelve thousand terms and 100 after an astronomically large number of terms. A series whose terms do not shrink to zero must diverge, because adding terms that stay sizeable cannot settle. A series whose terms shrink fast enough, like 1/n², converges, and its partial sums approach a limit, about 1.644934 for 1/n² summed forever, known as pi squared over six. The calculator’s convergence indication is based only on the numbers it computed: if the latest terms are very small, it says the series may be settling, and if the latest terms are not shrinking, it says divergence is consistent with what is seen. In both cases it labels the note as an indication, not a proof, because no finite computation can prove what happens at infinity, and the article repeats that warning deliberately.

The tool starts at n = 1, which is the standard convention for most textbook series, and it is worth noting when a series you meet starts elsewhere. A series starting at n = 0, like the exponential series 1 + x + x²/2! + x³/3! and so on, can be handled by shifting the formula: replace n by n − 1 in the term formula, or compute the first term separately and add it. The calculator does not do that shifting for you, and stating the convention plainly prevents the commonest off-by-one error in series work, where a student computes a sum that is missing its first term or includes an extra one and cannot see why the answer differs from the book’s.

How to Use the Calculator

Using the tool takes less than a minute. Follow these steps:

1. Enter the term formula a(n) using n, numbers, the operators + − * / ^, brackets, and the functions sin, cos, tan, exp, log, ln, sqrt, and abs, plus the constants pi and e. For example, type 1/n^2.
2. Enter the number of terms N, a whole number from 1 to 1000. The live test uses 10.
3. Press “Calculate Partial Sums”. The tool shows the partial sum after N terms, a convergence indication based on the terms computed, and the step-by-step terms and running totals.
4. Read the steps as a sequence: each line gives a term and the total so far, so you can see whether the totals are climbing quickly, slowly, or hardly at all.
5. If you want to explore further, increase N and watch how the total changes. A converging series will change less and less as N grows; a diverging series will keep changing, or keep growing.

A few typing habits will save you errors. Write multiplication explicitly: 1/n^2 is clear, but 1n or n2 will not be read as you intend. Use brackets around denominators and exponents: write 1/(n^2 + 1) rather than 1/n^2 + 1 when you mean the reciprocal of the sum, because the tool follows the usual order of operations and will read the unbracketed version differently. Remember that ^ means power, so 1/n^2 means one divided by n squared, and (-1)^n gives alternating signs, which is how alternating series are typed. And remember that the tool starts at n = 1, so a formula containing 1/(n − 1) will fail at the first term, with a clear error, rather than silently skipping it.

Worked Examples

Example 1: 1/n², N = 10, the live test

This is the test used to check the tool, and it is a historically famous series: the sum of the reciprocals of the squares, whose infinite sum is pi squared over six, about 1.644934, a result Euler found in 1734 and which stunned mathematicians of his day. With N = 10, the terms are 1, 0.25, 0.111111, 0.0625, 0.04, 0.027778, 0.020408, 0.015625, 0.012346, and 0.01, and the partial sums climb 1, 1.25, 1.361111, 1.423611, 1.463611, 1.491389, 1.511797, 1.527422, 1.539768, and finally 1.549768. The calculator returns exactly that final total, 1.549768, with every step shown. Comparing with the infinite sum, 1.644934, shows that ten terms capture most of the total but leave about 0.095 unaccounted for, in the tail of terms beyond ten. That tail is itself a useful lesson: convergence can be slow, and ten terms, while instructive, are not the same as infinity. With N = 100, the total is about 1.634984, much closer, and with N = 1000 it is about 1.644935, closer still, approaching 1.644934 from below, as it must, since every term is positive.

Example 2: The geometric series 1/2^n, N = 10

Take the term formula (1/2)^n, which gives 0.5, 0.25, 0.125, and so on, halving each time. The partial sums are 0.5, 0.75, 0.875, 0.9375, and after ten terms 0.9990234375, approaching 1 from below and never passing it. The calculator returns 0.999023 for N = 10, displayed to six decimal places. This series converges quickly, because its terms halve each time, and it is the classic example of a sum that is finite despite having infinitely many terms. It is also the series behind the old paradox of Achilles and the tortoise, and behind the practical fact that a repeating binary expansion like 0.111… in binary equals 1. The geometric series has a closed form, a/(1 − r) for first term a and ratio r with absolute value less than one, which here gives 0.5/(1 − 0.5) = 1, but the calculator does not use that formula; it adds the terms, so you can see the approach to 1 happening step by step rather than being told the limit.

Example 3: The harmonic series 1/n, N = 10 and beyond

Take the term formula 1/n, giving 1, 0.5, 0.333333, 0.25, and so on. After ten terms the partial sum is about 2.928968, and the calculator returns that total. The terms are shrinking, which might suggest convergence, but this series diverges, famously and slowly. After one hundred terms the sum is about 5.187, after one thousand terms about 7.485, and it grows without bound, roughly like the natural logarithm of N, though it takes an enormous N to reach even 100. This example is included because it is the single best antidote to the intuition that shrinking terms guarantee convergence. They do not. The calculator’s indication for N = 10 will note that the latest terms are small but not very small, and that no settling is visible, which is the correct cautious reading: ten terms of the harmonic series look like they might be settling, and they are not. Only a proof, such as grouping terms in blocks that each exceed a half, establishes divergence, and no finite computation can substitute for it.

Example 4: An alternating series, (-1)^(n+1)/n, N = 10

Take the term formula (-1)^(n+1)/n, which gives 1, −0.5, 0.333333, −0.25, 0.2, and so on, alternating in sign and shrinking in size. The partial sums oscillate around the eventual limit, which is the natural logarithm of 2, about 0.693147, approaching it from above and below alternately. After ten terms the sum is about 0.645635, and the calculator returns that total with its steps. Alternating series converge under a simple condition, Leibniz’s test: terms decrease in size towards zero, which these do, so the series converges, and the error after N terms is less than the size of the next term, which here is 1/11, about 0.0909, consistent with the actual error of about 0.0475. This error bound is one of the few cases where a finite computation does give a rigorous guarantee, and it is worth knowing, because it turns a partial sum into an approximation with a known worst-case error, rather than a mere guess.

Convergence Tests, and Why the Tool Does Not Claim Them

Textbooks teach a battery of convergence tests: the divergence test, the integral test, the comparison test, the ratio test, the root test, the alternating series test, and others. Each test gives a rigorous conclusion under stated conditions, and each requires reasoning about the term formula as a whole, not just about finitely many computed values. This calculator deliberately does not implement those tests as verdicts, because a verdict from finitely many terms would be dishonest: a series can behave one way for a thousand terms and another way thereafter, and a tool that announced “convergent” on the basis of a thousand terms would eventually be wrong. Instead, the tool gives an indication, labelled as such, based on whether the latest terms are shrinking and whether the partial sums appear to be settling, and it leaves the proof to the mathematics. That restraint is a feature, not a missing feature, and it is stated in the tool’s note, in this article, and in the publishing specification, so no reader is misled about what the tool can and cannot establish.

The ratio test deserves a brief mention, because it is the test most often applied to series with factorials and exponentials, and because its idea is easy to grasp: look at the ratio of successive terms, and if that ratio stays below one in the limit, the series converges, because eventually it behaves like a geometric series with ratio below one. The calculator does not compute that limit, but you can watch the ratios of its terms yourself: for 1/n², the ratio of term n+1 to term n approaches 1 from below, so the ratio test is inconclusive there, which is why 1/n² needs the integral test or a comparison to establish convergence. For (1/2)^n, the ratio is exactly 0.5, so the ratio test concludes convergence immediately. Knowing which test applies, and when a test is inconclusive, is part of learning series, and a tool that showed steps without claiming verdicts supports that learning better than a tool that announced verdicts it could not justify.

Limits: What This Tool Does and Does Not Do

This tool adds terms numerically, shows real partial sums step by step, and gives a labelled indication about whether the sums appear to be settling. It does not compute closed-form sums, does not prove convergence or divergence, and does not apply named convergence tests as verdicts. It starts at n = 1, accepts up to one thousand terms, and shows every step for N up to one hundred and the first twenty steps for larger N, with the final total always computed over all N terms. It rejects non-finite terms with a clear error rather than skipping them silently. It displays six decimal places, which is a display choice, not a guarantee of six correct decimals for every series; for series whose terms are computed accurately, it is comfortably accurate, and for series with cancellation between large positive and negative terms, floating-point error can accumulate, though for ordinary textbook series it is negligible. Within those limits, the step-by-step display lets you verify every total by adding the terms yourself, which is the point of a tool with “steps” in its name.

Privacy: Your Numbers Stay in Your Browser

Every number and formula you enter into this infinite series calculator stays in your browser. Nothing is uploaded to a server, nothing is stored, and there is no account to create. The calculation runs entirely on your own device. That matters even for a mathematics tool, because the formulas people enter can be part of coursework, an exam revision session, or a piece of unpublished work, and that work deserves to stay private. Closing the page removes your inputs, and no one else sees them. This browser-only design is the same principle behind the other free tools on Your Smart Kit: a simple tool that does one job well, keeps your data private, and gives an answer you can check.

Frequently Asked Questions

What is an infinite series?

It is a sum with infinitely many terms, defined by a term formula a(n) and studied through its partial sums, the sums of the first N terms. If the partial sums approach a fixed value as N grows, the series converges to that value; otherwise it diverges.

What is a partial sum?

It is the sum of the first N terms of a series, written S_N. The calculator shows partial sums step by step, so you can watch them grow, settle, or fail to settle, rather than being told only the final total.

Why does the calculator not tell me whether my series converges?

Because no finite computation can prove convergence or divergence. A series can behave one way for a thousand terms and another way thereafter. The tool gives a labelled indication from the numbers computed, and leaves proofs to convergence tests, which reason about the term formula as a whole.

What is the sum of 1/n²?

Summed to infinity, it is pi squared over six, about 1.644934, Euler’s famous result. Summed to ten terms, as in the live test, it is 1.549768, with the remaining tail of about 0.095 still to come in later terms.

Can a series converge if its terms do not tend to zero?

No. If the terms do not tend to zero, the series must diverge, because adding terms that stay sizeable cannot settle. That is the divergence test, and it is the one test that a finite computation can sometimes make obvious, when the terms are clearly not shrinking.

How many terms should I use?

Use enough that the total stops changing much when you add more terms, for a converging series, and use the step display to judge that. For 1/n², ten terms leave a tail of 0.095, one hundred terms leave about 0.01, and one thousand terms leave about 0.001, so the number of terms needed depends on the accuracy you want.

Related Tools on Your Smart Kit

If you found the infinite series calculator useful, you may also like the other free calculators on Your Smart Kit, including the piecewise integral calculator for integrating functions defined in pieces, the partial differentiation calculator for rates of change in two variables, and the sigma calculator for summing a finite list of terms. Each runs in your browser, free, with no sign-up, and each is built on the same principle: a simple tool that does one job well, keeps your data private, and gives an answer you can check. Try the infinite series calculator at the top of this page with your own term formula, and watch a sum that never ends settle, or fail to settle, one step at a time.