Key takeaways
- The Taylor series of eˣ is 1 + x + x²/2! + x³/3! + …
- More terms give a better answer, especially for x far from 0.
- This calculator counts the constant 1 as the first term, so 4 terms go up to x³/3!.
- The #1 mistake is forgetting the factorials.
In this guide
What is a Taylor series?
Some functions, like eˣ, sin x and cos x, are hard to work out directly. But adding up powers of x is easy. A Taylor series is an endless sum of powers that matches a function. If you stop after a few terms, you get a Taylor polynomial, a good approximation near the centre point.
Each term is chosen so the approximation has the same value, the same slope, the same bend, and so on, as the real function at the centre.
Why do Taylor series matter?
Taylor series turn hard functions into easy arithmetic. Scientists use the first two or three terms for quick estimates, like sin x ≈ x for small angles. Computers use series like these (along with other tricks) to work out eˣ, sin and cos.
With just 5 terms, the series gives e ≈ 2.7083. With 10 terms, it gives 2.71828, correct to five decimal places.
The formula
The general series around x = 0 is f(0) + f′(0)x + f″(0)x²/2! + …. Every derivative of eˣ is eˣ, and e⁰ = 1, so every coefficient is 1:
| Symbol | What it means | Example |
|---|---|---|
| x | The power you want to raise e to | 0.5 |
| n | The number of terms to add (1 to 20), counting the constant 1 as the first | 4 |
| k! | The factorial under the term xᵏ | 3! = 6 |
| Tₙ(x) | The approximation: the sum of the first n terms | ≈ 1.6458 |
The calculator also shows the true value of eˣ and the error, so you can see how good your approximation is. See the factorial calculator for factorials.
How to use a Taylor series step by step
- 1Write the terms. 1, x, x²/2, x³/6, x⁴/24, …
- 2Put in your x value. Work out each term.
- 3Add as many terms as you need. Each new term is smaller when x is small.
- 4Compare with the true value. The difference is the error.
Now try it yourself. Change the number of terms in the calculator below and watch the error shrink.
Try it yourself
Pre-filled with the example — change any value.
Common mistakes to avoid
- Using too few terms for big x. For x = 5, five terms give only 65.375, while e⁵ ≈ 148.41. The further you are from 0, the more terms you need.
- Miscounting terms. The constant 1 is the first term in this calculator. Some books count by the highest power instead, so “degree 4” means 5 terms here.
- Using degrees for sin and cos series. Those series only work with x in radians.
Tips and tricks
- Other famous series:
sin x = x − x³/3! + x⁵/5! − …andcos x = 1 − x²/2! + x⁴/4! − …. - Quick check: differentiate the series for eˣ term by term and you get the same series back. That matches the derivative of eˣ.
- Where terms come from: each coefficient uses a higher-order derivative of the function.
Where you’ll use it in real life
- Computing: software libraries use series and related methods to calculate eˣ, sin and cos.
- Physics: the small-angle rule sin θ ≈ θ makes pendulum maths simple.
- Finance: for small rates,
e^r ≈ 1 + r, a quick way to estimate continuous interest. - Engineering: simplifying hard formulas near a working point.
Quick summary
A Taylor series builds a function from powers of x. For eˣ, add 1 + x + x²/2! + x³/3! + … and stop when the answer is accurate enough. Use the calculator above to choose x and the number of terms, and see the true value and the error.
Worked example
Without a calculator’s eˣ button, you want to estimate e^0.5 (how much US $1 grows to after one year at 50% interest compounded continuously) using the first 4 terms of the series. Here x = 0.5 and n = 4.
- 01Tₙ(x) = Σₖ₌₀^3 x^k / k!
- 02x = 0.5, terms = 4
- 03k=0: 0.5^0/1 = 1
- 04k=1: 0.5^1/1 = 0.5
- 05k=2: 0.5^2/2 = 0.125
- 06k=3: 0.5^3/6 = 0.0208
- 07Approximation = 1.6458
- 08Exact eˣ = 1.6487, Error = 0.0029
These numbers are pre-filled in the calculator above.
Frequently asked questions
What is the difference between a Taylor series and a Maclaurin series?
A Maclaurin series is simply a Taylor series centred at x = 0. The eˣ series in this calculator is both.
How many terms of a Taylor series do I need?
It depends on how close x is to the centre and how accurate you need to be. For eˣ at x = 1, 10 terms give 5 correct decimal places.
Does the Taylor series of e^x work for every x?
Yes. The series for eˣ gets as close as you like to the true value for any x, including negative x, if you add enough terms.
