A series sum calculator adds up the terms produced by a formula. You supply a general term written in the index n, a starting index, and either a finishing index or the instruction to keep going forever. The tool evaluates every term, accumulates the total, and reports it. When the upper limit is infinity it does the only thing a numerical tool honestly can: it sums a very large number of terms, watches how the terms behave, and tells you whether the sequence of partial sums looks like it is settling on a value or running away.
Arb Digital publishes this alongside a bank of free mathematics tools because summation shows up far outside pure maths — in compound growth, in queueing models, in the discounted cash flow behind any subscription valuation. The arithmetic sequence calculator handles sums where each term rises by a constant amount and the geometric sequence calculator handles a constant ratio, both with closed-form shortcuts. This page is the general one: it takes any expression in n, including ones with no closed form at all, and grinds it out term by term.
What This Series Sum Calculator Does
It parses your expression with a small built-in mathematics parser, so 1/n^2, 3*n+1, (-1)^n/n, 1/fact(n) and ln(n)/n^3 all work. It then evaluates a(n) at every integer from your lower index to your upper index and adds the results in order. The headline figure is the sum. Below it you get the number of terms actually used, the first and last term values, and an estimate of the ratio-test limit.
That ratio-test figure is the interesting one. It is the numerical limit of the absolute value of a(n+1) divided by a(n) as n grows large. If it settles below 1 the series converges absolutely; above 1 it diverges; and if it sits at exactly 1 the ratio test is inconclusive and the tool says so rather than guessing. The panel underneath the numbers spells out that verdict in words, along with a tail estimate showing how much the sum is still moving over the final stretch of terms.
How to Use It
- Write the general term. Use n as the index and put explicit multiplication signs in.
2*nis understood,2nis not. - Set the lower index. Most textbook series start at n = 1, but geometric series are often written from n = 0. The tool respects whichever you choose.
- Choose finite or infinite. For a partial sum, pick a finite upper index and type it in. For an infinite sum, switch the selector and set how many terms to use for the approximation.
- Read the verdict, not just the number. The panel tells you whether an infinite total is meaningful for the expression you typed.
- Check the tail. If the last few thousand terms are still adding a visible amount, the estimate has not converged and you need more terms or an analytic method.
The Formula and How It's Calculated
A finite series is simply S = a(m) + a(m+1) + … + a(N) for a lower index m and an upper index N. There is no cleverness in the arithmetic; the only difficulty is evaluating the general term correctly, which is what the parser handles. An infinite series is defined as the limit of that partial sum as N grows without bound. It exists only when the partial sums approach a fixed value.
Work the default by hand. With a(n) = 1/n², from n = 1 to 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. Adding them gives 1.549768. Switch the upper index to infinity and the tool returns a value close to 1.644934, which is the famous exact value π²/6 — the Basel problem. The gap between the ten-term partial sum and the true limit is about 0.095, which is why a partial sum should never be quoted as an infinite total without saying how many terms went into it. The NIST Digital Library of Mathematical Functions section on the calculus of a complex variable sets out the absolute and uniform convergence criteria the tool applies.
Why an Infinite Sum Is Only Ever an Estimate Here
No numerical tool can add infinitely many numbers. What it can do is add a great many and inspect the trend. For a rapidly converging series such as 1/n! or 1/2ⁿ, a few dozen terms is enough to reach the limit of double-precision arithmetic, and the answer is exact for every practical purpose. For a slowly converging series it is a different story. The sum of 1/n² needs roughly a million terms to pin down six decimal places, because the tail beyond term N is about 1/N.
The harmonic series 1/n is the cautionary example. Its terms shrink towards zero, which looks reassuring, but the sum grows without bound — it just grows logarithmically, so slowly that no partial sum ever looks like it is diverging. Sum a million terms and you get about 14.39. Sum a billion and you get about 21.3. A calculator that quietly reported 14.39 as "the sum" would be lying. That is why this page always prints the convergence verdict beside the number, and why the ratio test result is one of the four headline figures rather than an afterthought.
Series With Closed Forms Worth Knowing
Some series collapse to a formula and never need a loop. A finite geometric series with first term a and ratio r sums to a(1 − r^N)/(1 − r), and the infinite version converges to a/(1 − r) whenever the absolute value of r is less than one. A finite arithmetic series sums to N times the average of the first and last terms — the trick attributed to a young Gauss. The sum of the first N whole numbers is N(N+1)/2, the sum of their squares is N(N+1)(2N+1)/6, and the sum of their cubes is the square of the sum of the numbers themselves.
Type any of those into this tool and you will get the same answer the formula gives, which is a good way to check that you have transcribed the formula correctly. Where the closed forms earn their keep is in symbolic work and in cases where N is astronomically large. Paul Dawkins' Calculus II series and sequences notes at Lamar University derive these results in full and cover the standard convergence tests in the order most courses teach them.
Alternating Series and the Order You Add In
An alternating series such as 1 − 1/2 + 1/3 − 1/4 + … converges, to the natural logarithm of 2, even though the same terms without the alternating signs diverge. That is called conditional convergence, and it has a genuinely startling consequence: the sum of a conditionally convergent series depends on the order you add the terms in. Rearrange them and you can make the total come out to any real number you like. This is Riemann's rearrangement theorem, and it is the reason careful texts insist on absolute convergence before allowing any regrouping.
Practically, it means that if your series has mixed signs and the absolute values of the terms sum to infinity, the number this tool gives you is the sum in the order written, which is the only order that has a defensible claim to being the answer. The tool computes the ratio test on absolute values, so a conditionally convergent series will show a limit of 1 and an inconclusive verdict — correctly, because the ratio test cannot distinguish these cases.
Floating Point and the Order of Accumulation
Adding many small numbers to one large running total loses precision, because each addition rounds to the nearest representable double. Sum a million copies of 0.1 by naive accumulation and the answer drifts from 100,000 by a small but real amount. This tool accumulates in index order, which is what a textbook means by the partial sum, and it is accurate to roughly twelve significant figures for the term counts allowed here.
If you need more than that, the standard remedy is to sum the smallest terms first, which is the reverse of index order for a decreasing series, or to use a compensated summation scheme that tracks the rounding error separately. Neither changes the mathematics; both change the last few digits. It is worth knowing the difference exists before you conclude that two tools disagree about a sum when they actually agree to eleven places and differ in the twelfth. Our scientific notation converter is useful when a term outgrows what ordinary decimal display can show.
Where Series Turn Up Outside a Maths Class
The present value of a perpetuity is an infinite geometric series. The expected number of trials before a first success is a series in disguise. The total lifetime revenue from a subscriber with a fixed monthly churn rate is a geometric series whose ratio is the retention rate, which is exactly why lifetime value equals monthly margin divided by churn. Compound interest schedules, radioactive decay chains, and the settling behaviour of a control system are all series problems wearing different clothes.
Marketing analytics runs into them constantly. Attribution decay models, frequency capping, and cohort revenue curves all reduce to summing a term formula over a horizon. When the horizon is genuinely long, the infinite sum is often a cleaner and more defensible number than an arbitrary cut-off at 24 months, provided the ratio is safely below one. Our compound interest calculator and percentage difference calculator handle the everyday versions of those calculations directly.
Arb Digital builds the measurement and forecasting layer behind growth programmes, so the numbers in the board deck survive being checked.
Browse All Free Tools Talk To Our TeamCommon Mistakes to Avoid
- Writing 2n instead of 2*n — the parser needs explicit multiplication, and an unrecognised term produces an error rather than a silent wrong answer.
- Starting at the wrong index — a geometric series written from n = 0 and the same one written from n = 1 differ by a whole term.
- Quoting a partial sum as an infinite sum without stating how many terms were used.
- Assuming that terms shrinking to zero means the series converges — the harmonic series is the standing counterexample.
- Rearranging a conditionally convergent series to make it easier to add, which can change the answer entirely.
Related Free Tools From Arb Digital
For a constant difference between terms use the arithmetic sequence calculator, and for a constant ratio use the geometric sequence calculator, both of which give the nth term as well as the sum. If you are summing squared deviations rather than a term formula, the sum of squares calculator is the right page. Expand a logarithmic term before summing with the logarithm expansion calculator, or browse everything on the free online tools hub.
Frequently Asked Questions
A sequence is the list of terms produced by the formula. A series is what you get when you add those terms together. This tool takes the sequence formula and returns the series total.
No. It sums a very large number of terms and reports the result along with a convergence verdict and a tail estimate. For rapidly converging series that estimate reaches the limit of double-precision arithmetic; for slow ones it does not.
The tool applies the ratio test numerically. A limit below one means absolute convergence, above one means divergence, and exactly one means the test is inconclusive and another test is needed.
Terms shrinking to zero is necessary for convergence but not sufficient. The terms of the harmonic series shrink too slowly, so the partial sums grow without bound, though only at a logarithmic rate.
It means the test gives no information. Series with a ratio limit of one include both convergent cases such as the sum of one over n squared and divergent cases such as the harmonic series.
For an absolutely convergent series, no. For a conditionally convergent series it matters completely, because rearranging the terms can change the total to any value you choose.
Addition, subtraction, multiplication, division, powers and brackets, plus sqrt, ln, log, exp, abs, sin, cos and fact for the factorial. The index variable must be written as n.
This tool is provided for educational use. Numerical summation is subject to floating-point rounding, and an approximated infinite sum should not be treated as a proof of a closed-form value.
