Percentage
A percentage is a number indicated as a fraction of 100. It is a dimensionless number often expressed using the symbol %.
Algebraic Expressions
In mathematics, an algebraic expression consists of constant(s), variable(s), and mathematical operators. It is made up of terms.
Numbers
Numbers are some measures used for counting. They can be compared one with another to know its position in the number line and determine which one is greater or lesser than the other.
Subtraction
Before we begin to understand the subtraction of algebraic expressions, we need to list out a few things that form the basis of algebra.
Addition
Before we begin to understand the addition of algebraic expressions, we need to list out a few things that form the basis of algebra.
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![## Taylor Polynomial Expansion
### Problem Statement:
Find the Taylor polynomial centered at \( a = 0 \) for each function. Use summation notation or expanded notation.
Given function:
\[ f(x) = e^{(5x)} \]
### Detailed Steps:
To determine the Taylor polynomial for \( f(x) = e^{5x} \) centered at \( a = 0 \), we use the Taylor series expansion formula:
\[ f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x - a)^n \]
Since \( a = 0 \), the formula simplifies to:
\[ f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n \]
The function \( f(x) = e^{5x} \) has the property that its derivatives are:
\[ f^{(n)}(x) = \frac{d^n}{dx^n} e^{5x} = 5^n e^{5x} \]
Evaluating at \( x = 0 \), we get:
\[ f^{(n)}(0) = 5^n \]
Plugging this into the Taylor series formula yields:
\[ f(x) = \sum_{n=0}^{\infty} \frac{5^n}{n!} x^n \]
Or in expanded form:
\[ f(x) = 1 + 5x + \frac{(5x)^2}{2!} + \frac{(5x)^3}{3!} + \frac{(5x)^4}{4!} + \cdots \]
In summary, the Taylor polynomial for \( f(x) = e^{5x} \) centered at \( a = 0 \) is:
\[ f(x) = \sum_{n=0}^{\infty} \frac{5^n}{n!} x^n \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa7c87654-3157-4119-9345-56717eaa1677%2F04fcbb44-008a-4eda-a31a-a838d090f46a%2F3ceh0rs_processed.png&w=3840&q=75)
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