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Taylor series of e⁻ˣ up to degree 3

The Taylor polynomial of degree 3 for e⁻ˣ about 0 (the Maclaurin polynomial) is 1 - x + x²/2 - x³/6. The full series converges for every x.

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e⁻ˣ and its degree 3 Taylor polynomial-3-2-11235101520e⁻ˣdegree 3
e⁻ˣ and its degree 3 Taylor polynomial. The polynomial follows the curve closely near 0 and drifts away further out; more terms stretch the good region.

How to work it out

  1. 1

    The Taylor series of f about 0 uses the derivatives of f at 0:

    f(x)=∑k=0∞f(k)(0)k! xkf(x) = \sum_{k=0}^{\infty} \frac{f^{(k)}(0)}{k!}\,x^k
  2. 2

    For f(x) = e⁻ˣ the first derivatives at 0 are:

    f(0)(0)=1,f(1)(0)=−1,f(2)(0)=1,f(3)(0)=−1f^{(0)}(0) = 1,\quad f^{(1)}(0) = -1,\quad f^{(2)}(0) = 1,\quad f^{(3)}(0) = -1
  3. 3

    Divide each by k! and keep the terms up to degree 3:

    e−x≈1−x+12x2−16x3e^{-x} \approx 1 - x + \frac{1}{2}x^{2} - \frac{1}{6}x^{3}
  4. 4

    Radius of convergence: ∞. Inside it, more terms give a better approximation; outside, the series does not settle.

LaTeX source of the formulas on this page
\[ e^{-x} \approx 1 - x + \frac{1}{2}x^{2} - \frac{1}{6}x^{3} \]
\[ f(x) = \sum_{k=0}^{\infty} \frac{f^{(k)}(0)}{k!}\,x^k \]
\[ f^{(0)}(0) = 1,\quad f^{(1)}(0) = -1,\quad f^{(2)}(0) = 1,\quad f^{(3)}(0) = -1 \]

Degree 3 polynomial against the real e⁻ˣ

xPolynomialExactError
0.10.904833330.904837424.08e-6
0.50.604166670.606530662.36e-3
0.90.383500000.406569662.31e-2
2-0.333333330.135335284.69e-1

Questions

What is the Taylor series of e⁻ˣ?

1 - x + x²/2 - x³/6 + x⁴/24 - x⁵/120 + x⁶/720 - x⁷/5040 + x⁸/40320 + …

What is the degree 3 Taylor polynomial of e⁻ˣ?

1 - x + x²/2 - x³/6

What is the radius of convergence of the series of e⁻ˣ?

∞

What is the Maclaurin series?

A Taylor series centred at 0.

Other degrees of e⁻ˣ

Degree 3 of other functions

    Taylor series of e⁻ˣ: degree 3 polynomial and error | WordGPT