Series Sum
Sum mathematical series from any start to end value. Supports expressions with n, exponents, and common mathematical functions.
The starting value of the summation index n
The ending value of the summation index n
Write the formula using n as the variable. Use ^ for exponents. Supports pi for π.
The formula
S = Σ_{n=start}^{end} f(n)
- n
- Summation Index
- start
- Lower Bound
- end
- Upper Bound
- f(n)
- Expression
- S
- Sum
How Series Sum Works
S = Σ_{n=start}^{end} f(n)
The series calculator computes the sum of a mathematical sequence from a start index to an end index. For each integer value of n from start to end, the expression f(n) is evaluated and all results are added together. This is denoted by the capital Greek letter Sigma (Σ), which represents summation. The lower bound (n = start) and upper bound (end) define the range of summation.
- n
- Summation Index — The integer variable that takes each value from start to end during summation.
- start
- Lower Bound — The first integer value of n at which the expression is evaluated.
- end
- Upper Bound — The last integer value of n at which the expression is evaluated.
- f(n)
- Expression — The mathematical formula evaluated at each n, defining the sequence terms.
- S
- Sum — The total sum of all terms from n = start to n = end.
Worked Examples
A student calculates the sum of the first 10 natural numbers to verify the well-known formula n(n+1)/2. They use the expression "n" from n=1 to n=10.
start = 1 · end = 10 · expression = n
Sum = 55. First 20 terms: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. Number of terms: 10. Last term: 10. Matches the arithmetic series formula n(a1 + an)/2 = 10(1+10)/2 = 55.
The series computes 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 55. This matches the arithmetic series formula: sum = n(a1 + an)/2 = 10(1 + 10)/2 = 55. The sum of the first n natural numbers is a classic result that appears in combinatorics, statistics (sum of ranks), and algorithm analysis of simple nested loops.
A physics student computes the partial sum of the harmonic series 1/n from n=1 to n=100 to understand how slowly this series diverges, comparing it to the natural log approximation H_n ≈ ln(n) + γ.
start = 1 · end = 100 · expression = 1/n
Sum ≈ 5.1873775176 (harmonic number H_100). First 20 terms: 1, 0.5, 0.3333333333, 0.25, 0.2, 0.1666666667, 0.1428571429, 0.125, 0.1111111111, 0.1, ... Number of terms: 100. Last term ≈ 0.01. ln(100) + γ ≈ 5.182, a difference of only ~0.005.
The sum of 1/n from 1 to 100 is approximately 5.187. For comparison, ln(100) + γ ≈ 4.605 + 0.577 = 5.182, a difference of only 0.005 — demonstrating the accuracy of the Euler-Mascheroni approximation even for moderate n. The harmonic series diverges, but extremely slowly: to exceed a sum of 10, you need over 12,000 terms.
Understanding the Result
A series is the sum of the terms of a sequence. The notation Σ_{n=a}^{b} f(n) means "evaluate the expression f(n) for each integer n starting at a, ending at b, and add all the results together." Series are fundamental in mathematics and appear throughout calculus, number theory, and applied mathematics. Finite series have a definite number of terms and always yield a finite sum. Common examples include arithmetic series (where consecutive terms differ by a constant), geometric series (where consecutive terms have a constant ratio), and harmonic series (terms of the form 1/n).
The series calculator can handle any expression using standard arithmetic operations, exponents, and the constant π. By varying the expression and bounds, you can explore a wide range of mathematical series — from simple sums of consecutive integers to more complex sequences involving powers and reciprocals.
How to Use
- Enter the starting value (n start) and ending value (n end) for the summation index.
- Write the expression using n as the variable. Use ^ for exponents (e.g., 2^n, n^2).
- Use pi for the mathematical constant π. The calculator supports basic arithmetic operators and parentheses.
- View the total sum, the first 20 terms in the series, the number of terms, and the last term value.
- The expression is safely evaluated for each integer n in the range. If any term cannot be evaluated (for example 1/(n-5) at n = 5) the calculator tells you which n failed instead of skipping it.
Quick Reference
| Sum of constants | Σ c = c × (b − a + 1) |
| Sum of n | Σ n = b(b+1)/2 − (a−1)a/2 |
| Geometric series | Σ r^n = (r^(b+1) − r^a)/(r − 1) |
| Harmonic terms | Σ 1/n diverges as n → ∞ |
Frequently Asked Questions
- What types of expressions does the series calculator support?
- The calculator supports basic arithmetic operations (+, -, *, /), exponents (^), parentheses for grouping, the constant pi (π), and the variable n. Examples: 1/n, 1/2^n, n^2, 1/(n*(n+1)), n/(n+1). Functions such as sqrt, sin, cos, tan, ln, log, abs, exp and the factorial operator ! are also available. Terms are double-precision numbers (the running sum uses compensated addition), so very large values may have floating-point precision limitations.
- What happens if the start value is larger than the end value?
- If the start value exceeds the end value, the calculator shows a message instead of a result. The summation requires a valid range where start ≤ end. If you need a descending sum, reverse the start and end values and negate the expression if necessary.
- Does the calculator support infinite series?
- No, this calculator computes finite sums from a specified start to a specified end value. To approximate an infinite series, you can choose a large end value where additional terms become negligible. The calculator shows the number of terms summed, helping you gauge how close you might be to the infinite limit.
- What is the difference between a sequence and a series?
- A sequence is an ordered list of numbers: a₁, a₂, a₃, ... A series is the sum of those numbers: a₁ + a₂ + a₃ + ... = Σ a_n. In this calculator, the sequence is defined by f(n) evaluated at each integer, and the series is the cumulative sum of those sequence values.
- Why are only the first 20 terms shown in the result?
- For readability and performance, the calculator displays the first 20 terms of the series as a comma-separated list. All terms are still included in the total sum calculation. The "Number of Terms" result tells you exactly how many terms were summed, and the "Last Term Value" shows the final term in the range.
Common Uses
- •Computing the total accumulated value of a sequence, such as the sum of the first N integers or squares.
- •Evaluating finite approximations of infinite series to estimate the value of convergent series.
- •Calculating partial sums of arithmetic and geometric progressions in financial mathematics.
- •Analyzing the behavior of series by examining how partial sums grow as more terms are added.
- •Computing discrete approximations of integrals using Riemann sums where f(n) represents sampled values.
Pro Tips
- →Start and end must be whole numbers. Use * for multiplication (write 2*n, not 2n), ^ for powers, ! for factorials, and functions such as sqrt, sin, cos, ln, log, abs, exp.
- →Terms are computed as double-precision numbers (up to about 10^308); if a term or the sum overflows, the calculator reports it rather than skipping it.
- →For an arithmetic series with first term a and difference d use the expression a + (n - 1)*d from n = 1; for a geometric series use a*r^(n - 1).
- →For alternating series, use expressions like (-1)^n / n for the alternating harmonic series, or (-1)^n / (2*n+1) from n = 0 for the Leibniz series, whose sum tends to pi/4.
- →The calculator sums at most 10,000 terms. If your range is longer, only the first 10,000 terms are summed and a "Range shortened" note shows the end index actually used.
- →To explore convergence behavior, try a series known to converge (like 1/2^n) with increasing end values and observe how the sum approaches its limit — the "Last Term Value" shows when terms become negligible.
Limitations to Know
- •Limited to 10,000 terms (start + 9,999) to keep the page responsive. A longer range is shortened, with a note showing the end index that was actually summed.
- •A term or sum beyond about 10^308 overflows a double; the calculator reports the first n where this happens rather than undercounting the total.
- •Expressions that are undefined inside the range (like 1/(n-5) when n = 5) stop the calculation with a message naming the failing n, so a wrong total is never shown.
- •Finite-precision floating-point arithmetic (IEEE 754 double) means sums of many small terms or terms with very different magnitudes may lose precision due to rounding accumulation. For high-precision summation needs, use a dedicated CAS.
- •This calculator computes finite sums only and cannot evaluate infinite series limits or determine convergence/divergence. Use the Taylor Series calculator for series expansions and convergence analysis.
Cite this calculator
TheCalcUniverse. "Series Calculator — Sum a Series from n Start to End." TheCalcUniverse, 2026, https://thecalcuniverse.com/math/series/. Accessed October 25, 2026.
Embed this calculator on your site
You may also like
-
Mathematics
Taylor Series
Compute Taylor series of common functions like e^x, sin(x), cos(x) around any center point. Adjust the expansion order for accurate polynomial approximation.
Open calculator → -
Mathematics
Limit Calc
Compute limits of functions as x approaches any value. Evaluate left-hand, right-hand, and two-sided limits with approach tables.
Open calculator → -
Mathematics
Number Sequences
Identify arithmetic, geometric, and Fibonacci sequences. Find common difference, common ratio, nth-term formula, and compute next terms.
Open calculator → -
Mathematics
Integral Calc
Calculate definite integrals numerically using Simpsons rule with step-by-step formula breakdown. Supports polynomial, trig, and exponential functions.
Open calculator →