Question

Write out the sum of the first 5 terms of the given power series. $$\sum_{n=0}^{\infty} \frac{1}{n !} x^{n}$$

Short answer

The sum is \( 1 + x + \frac{x^2}{2} + \frac{x^3}{6} + \frac{x^4}{24} \).

Step-by-step solution

Understand the Task

We need to find the sum of the first 5 terms of a given power series. The series given is \( \sum_{n=0}^{\infty} \frac{1}{n!} x^n \). This means we will sum the terms from \( n=0 \) to \( n=4 \).

Write Out the General Term

The general term of the series is \( \frac{1}{n!} x^n \). We will substitute values of \( n \) from 0 to 4 into this general term to get the first five terms.

Calculate Each Term

For each \( n \) from 0 to 4, calculate the term:- For \( n = 0 \), the term is \( \frac{1}{0!} x^0 = 1 \). - For \( n = 1 \), the term is \( \frac{1}{1!} x^1 = x \).- For \( n = 2 \), the term is \( \frac{1}{2!} x^2 = \frac{x^2}{2} \).- For \( n = 3 \), the term is \( \frac{1}{3!} x^3 = \frac{x^3}{6} \).- For \( n = 4 \), the term is \( \frac{1}{4!} x^4 = \frac{x^4}{24} \).

Sum First Five Terms

Add the terms from \( n=0 \) to \( n=4 \) to get:\[ 1 + x + \frac{x^2}{2} + \frac{x^3}{6} + \frac{x^4}{24} \].

Final Answer

Thus, the sum of the first five terms of the series is: \[ 1 + x + \frac{x^2}{2} + \frac{x^3}{6} + \frac{x^4}{24} \].

Key concepts

Sum of Series

To understand power series better, it's essential to become familiar with the concept of the "sum of series." A series is a way of expressing the sum of an infinite sequence of numbers. When we talk about the sum of a series, especially one that is infinite, we often mean finding a partial sum. This means just taking the first few terms and adding them up.

In our case, the task is to find the sum of the first 5 terms of a power series. This kind of series is expressed in the form \( \sum_{n=0}^{\infty} \frac{1}{n!} x^n \). For finding the partial sum, the first few terms \( \left(n = 0 \text{ to } 4 \right)\) are calculated, and their results are then added together. This gives an approximation of the series, which can be particularly insightful for understanding its behavior without considering all its infinite terms.

Factorials

Factorials may seem intimidating at first, but they are quite simple once you understand the concept. Denoted as \(n!\), a factorial of a non-negative integer \(n\) is the product of all positive integers less than or equal to \(n\).

Here are some quick examples to remember:

• \(0! = 1\) - by convention, the factorial of zero is defined as 1.
• \(1! = 1\)
• \(2! = 2 \times 1 = 2\)
• \(3! = 3 \times 2 \times 1 = 6\)
• \(4! = 4 \times 3 \times 2 \times 1 = 24\)

Factorials show up in many areas of mathematics, often related to series like ours, primarily because they serve as an important mathematical tool for organizing terms in sequences.

Exponential Function

The exponential function is a crucial concept in mathematics, especially when discussing power series. It is often expressed as \(e^x\) or \(\exp(x)\). In fact, a very famous representation of the exponential function is through an infinite series: \[ e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} \] The series form helps us approximate \(e^x\) by summing its initial terms. Interestingly, the power series \(\sum_{n=0}^{\infty} \frac{1}{n!} x^n\) mimics this pattern closely.

By summing up the first few terms of the series, you get increasingly precise approximations of the exponential function. It is a testament to how series and exponential functions are deeply intertwined in both theoretical and practical applications.

Mathematical Induction

Mathematical induction is a powerful method used to prove statements or formulas in mathematics, especially when the statement is about an infinite number of cases. The principle of induction comprises of two primary steps: proving a base case and proving the induction step.

The base case is the simplest instance of the statement, often starting from the smallest number, like \(n = 0\) or \(n = 1\). Once the base case is proven, the induction step involves assuming the statement is true for some arbitrary case \(n=k\), and then proving it true for \(n=k+1\).

Though not directly applied in simply summing a few terms of a series, understanding induction helps appreciate how mathematicians could prove properties or behaviors of infinite structures, such as our power series, thus ensuring that approximations and calculations align with theoretical expectations.