Taylor Series Calculator

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Example

Created on 2024-06-20Asked by Mason Martin (Solvelet student)

Find the Taylor series of

f(x) = e^x

centered at

x = 0

Solution

To find the Taylor series of

f(x) = e^x

centered at

x = 0

: 1. **Formula for the Taylor series:**

f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x - a)^n.

2. **Evaluate the derivatives at

a = 0

:**

f^{(n)}(0) = e^0 = 1 \quad \forall n \in \mathbb{N}.

3. **Form the series:**

e^x = \sum_{n=0}^{\infty} \frac{1}{n!} x^n.

4. **Result:** The Taylor series of

e^x

centered at

x = 0

is:

e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \cdots.

Solved on Solvelet with Basic AI Model

Some of the related questions asked by Camila Perez on Solvelet

1. Find the Maclaurin series representation of the function

f(x)=\sin(x)

.2. Use the Taylor series for

e^x

to approximate the value of

e^{0.5}

DefinitionA Taylor series is an infinite sum of terms calculated from the values of the derivatives of a function at a single point. It estimates functions in the neighborhood of that point. E.g.: The Taylor series to ex at x=0 is 1+x+2! x2+3! x3+⋯.

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