Question

A series \(\sum_{n=1}^{\infty} a_{n}\) is given. (a) Give the first 5 partial sums of the series. (b) Give a graph of the first 5 terms of \(a_{n}\) and \(S_{n}\) on the same axes. $$\sum_{n=1}^{\infty} \frac{1}{n !}$$

Short answer

First 5 partial sums: 1, \( \frac{3}{2} \), \( \frac{4}{3} \), \( \frac{61}{24} \), \( \frac{63}{24} \); plot terms and sums on the same axes.

Step-by-step solution

Understanding the Given Series

The series provided is \( \sum_{n=1}^{\infty} \frac{1}{n!} \). It's an infinite series where each term is defined as \( a_n = \frac{1}{n!} \). Our task is to find the first 5 partial sums of this series and plot both the terms and partial sums.

Calculating First 5 Terms

Calculate the first 5 terms of the series \( a_n \):- \( a_1 = \frac{1}{1!} = 1 \)- \( a_2 = \frac{1}{2!} = \frac{1}{2} \)- \( a_3 = \frac{1}{3!} = \frac{1}{6} \)- \( a_4 = \frac{1}{4!} = \frac{1}{24} \)- \( a_5 = \frac{1}{5!} = \frac{1}{120} \)

Calculating the First 5 Partial Sums

Calculate the partial sums \( S_n = \sum_{i=1}^{n} a_i \):- \( S_1 = a_1 = 1 \)- \( S_2 = a_1 + a_2 = 1 + \frac{1}{2} = \frac{3}{2} \)- \( S_3 = a_1 + a_2 + a_3 = 1 + \frac{1}{2} + \frac{1}{6} = \frac{8}{6} = \frac{4}{3} \)- \( S_4 = a_1 + a_2 + a_3 + a_4 = 1 + \frac{1}{2} + \frac{1}{6} + \frac{1}{24} = \frac{61}{24} \)- \( S_5 = a_1 + a_2 + a_3 + a_4 + a_5 = 1 + \frac{1}{2} + \frac{1}{6} + \frac{1}{24} + \frac{1}{120} = \frac{63}{24} \)

Graphing

Plot the terms \( a_n \) and partial sums \( S_n \) on the same axes:- Points for \( a_n \): (1, 1), (2, \( \frac{1}{2} \)), (3, \( \frac{1}{6} \)), (4, \( \frac{1}{24} \)), (5, \( \frac{1}{120} \))- Points for \( S_n \): (1, 1), (2, \( \frac{3}{2} \)), (3, \( \frac{4}{3} \)), (4, \( \frac{61}{24} \)), (5, \( \frac{63}{24} \))Draw a graph with \( n \) on the x-axis and the values on the y-axis, displaying how both sets of points trend.

Key concepts

partial sums

In an infinite series, a partial sum refers to the sum of the first few terms of the series. It gives us an idea of how the series behaves as we add more terms. The partial sum is especially useful in understanding convergence.
When tasked with calculating the first few partial sums of a series like\( \sum_{n=1}^{\infty} \frac{1}{n!}\),we start by adding up the initial terms one by one.

• \( S_1 = a_1 = 1 \)
• \( S_2 = a_1 + a_2 = 1 + \frac{1}{2} = \frac{3}{2} \)
• \( S_3 = a_1 + a_2 + a_3 = 1 + \frac{1}{2} + \frac{1}{6} = \frac{8}{6} = \frac{4}{3} \)
• \( S_4 = a_1 + a_2 + a_3 + a_4 = 1 + \frac{1}{2} + \frac{1}{6} + \frac{1}{24} = \frac{61}{24} \)
• \( S_5 = a_1 + a_2 + a_3 + a_4 + a_5 = 1 + \frac{1}{2} + \frac{1}{6} + \frac{1}{24} + \frac{1}{120} = \frac{63}{24} \)

Each partial sum \( S_n \) represents the sum of the first \( n \) terms of the series, helping us judge the trend of the series. By observing these sums, we can predict how the series grows and assess if it converges.

factorial

Factorials are a crucial concept in mathematics. The notation \( n! \) represents the product of an integer \( n \) with all the positive integers less than it. Essentially, \( n! = n \times (n-1) \times (n-2) \times \ldots \times 1 \).
For example:

• \( 3! = 3 \times 2 \times 1 = 6 \)
• \( 4! = 4 \times 3 \times 2 \times 1 = 24 \)
• \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \)

The series \( \sum_{n=1}^{\infty} \frac{1}{n!} \) features terminology involving factorials as the denominator for each term. As \( n \) increases, \( n! \) grows very large, which causes each succeeding fraction in the series \( \frac{1}{n!} \) to become very small.

graphing terms

Graphing is a visual method to analyze mathematical functions and series. For our series \( \sum_{n=1}^{\infty} \frac{1}{n!} \), plotting the terms \( a_n = \frac{1}{n!} \) and the partial sums \( S_n \) helps us see the trends.
To graph these:

• Plot \( a_n \) as points at \( (n, a_n) \) for each term: for example \( (1, 1), (2, \frac{1}{2}), (3, \frac{1}{6}) \).
• Plot \( S_n \) similarly: \( (1, 1), (2, \frac{3}{2}), (3, \frac{4}{3}) \).

By placing these on the same graph, you observe how quickly the terms decrease, reflecting their factorial denominator. Meanwhile, the partial sums provide a cumulative view, slowly increasing to a particular limit. Such a graph reveals convergence tendencies and the diminishing impact of higher-order terms.

convergence of series

Convergence is a key topic when studying infinite series. A series converges if its terms approach a fixed value, allowing for a meaningful sum.
In the context of \( \sum_{n=1}^{\infty} \frac{1}{n!} \), each term gets smaller as \( n \) increases, greatly influenced by the factorial in the denominator. Given that the factorial grows rapidly, this implies the terms of the series become negligible very fast.
As a result, the first few partial sums already give a good approximation of the series' total.Convergence is investigated by calculating these partial sums and observing their behavior; in this series, they tend to a finite number. That means the infinite sum of the series approaches a specific value.
This series is actually closely related to the value of \( e \), the base of natural logarithms, showing its convergence around \( e-1 \). Understanding convergence helps grasp the idea of infinite processes leading to definite results.