Unitary Method
The word “unitary” comes from the word “unit”, which means a single and complete entity. In this method, we find the value of a unit product from the given number of products, and then we solve for the other number of products.
Speed, Time, and Distance
Imagine you and 3 of your friends are planning to go to the playground at 6 in the evening. Your house is one mile away from the playground and one of your friends named Jim must start at 5 pm to reach the playground by walk. The other two friends are 3 miles away.
Profit and Loss
The amount earned or lost on the sale of one or more items is referred to as the profit or loss on that item.
Units and Measurements
Measurements and comparisons are the foundation of science and engineering. We, therefore, need rules that tell us how things are measured and compared. For these measurements and comparisons, we perform certain experiments, and we will need the experiments to set up the devices.
#9
![**Power Series and Taylor Series Problems**
**Problem 9:**
Find a power series for the function and determine the interval of convergence.
\[ f(x) = \frac{1}{1 - 3x} \]
**Problem 10:**
Find the Taylor series for the function, centered at \( c \), and determine the interval of convergence.
\[ f(x) = e^{-3x}, \quad \text{where } c = -2 \]
**Explanation:**
In problem 9, you are asked to express the given function \[ f(x) = \frac{1}{1 - 3x} \] as a power series. Recall that
\[ \frac{1}{1 - u} = \sum_{n=0}^{\infty} u^n \]
for \( |u| < 1 \). In this case, \( u = 3x \). Use this result to write the function as a series and find the interval of convergence where \( |3x| < 1 \).
In problem 10, you need to find the Taylor series expansion of the function \( f(x) = e^{-3x} \) centered at \( c = -2 \). Use the formula for the Taylor series:
\[ f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(c)}{n!} (x - c)^n. \]
Calculate the derivatives of \( f(x) \) evaluated at \( c = -2 \), and then substitute into the series formula to find the Taylor series and its interval of convergence.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F18c962eb-f527-44ff-b827-d97e28fe335c%2F65e9ce30-d311-4808-bba6-0579b640ef40%2Fj0a12o_processed.png&w=3840&q=75)
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