E Series Calculator
Euler e approximation and error from terms.
What it does
Euler's number hides in factorials upside down: one over zero factorial plus one over one plus one over two, racing to 2.71828 in a dozen terms. Bankers and biologists both spend it.
The tool above takes the term count. It returns the approximation, the error, and the count.
The default 10 terms gives 2.7182815256, off under 0.000001. The 5 case gives 2.7083333333 off 0.0099485.
Calculus texts carry this exponential series with factorial denominators and shrinking remainders. LibreTexts and OpenStax walk the same expansion.
Factorial speed humbles alternating crawls: e converges where pi plods. Float precision still ceilings every run near 15 digits.
How to use
- Type the term count.
- Read the approximation first.
- Read the error beside it.
- Read the term count last.
- Compare counts for digits.
- Respect float ceilings.
Formula
- n
- Term count, 1 to 200.
- approx
- Reciprocal factorial sum.
- error
- Distance from Math.E.
LibreTexts calculus carries the exponential series with factorial denominators and remainder behavior. OpenStax calculus mirrors the same expansion for e. Floating point caps precision near 15 digits; terms cap at 200 for page speed.
Worked example
Example: 10 terms of e
Default: 10 terms.
- Terms: 10 reciprocal factorials.
- Sum: 1 + 1 + 1/2 + ... + 1/9!.
- Approx: 2.7182815256.
- Error: under 0.000001 vs true e.
- Terms: 10 terms run.
- Tool shows 2.7182815256, under 0.000001, 10 terms.
Reference table
| Terms | Approx | Error |
|---|---|---|
| 5 | 2.7083333333 | 0.0099485 |
| 10 | 2.7182815256 | under 0.000001 |
| 100 | 2.7182818285 | under 0.000001 |
| 1 | 1 | 1.71828183 |
Limits
- Terms 1 to 200.
- Float precision ceilings.
- Series for e only.
- Display rounds at 10.
Common mistakes
- Counting from 1 not 0.
- Expecting pi from e terms.
- Ignoring float ceilings.
- Stopping at 2 terms blindly.
- Reading rounded tails exact.
Try it yourself
Quick checks. Answers sit tucked under each question.
5 terms?
2.7083333333 with error 0.0099485 in 5 terms.
100 terms?
2.7182818285 with error under 0.000001 in 100 terms.
0 terms?
Zero terms sum nothing: enter 1 to 200.
Questions
E series?
Sum of 1/k! from 0 up. Factorials explode, so terms shrink fast.
Converge how fast?
Very fast, factorial fast. Ten terms already show 9 digits.
Small counts error?
Yes, near 0.0099485 at 5 terms. Shown beside every run.
Huge counts exact?
No, floating point rounds past 15 digits. Big counts show under 0.000001.
Zero factorial?
Yes, 0! is 1, so the series opens 1 + 1 + ....
Pi too?
No, this sums e. Pi needs its own alternating series page.