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Chapter 5: Problem 71

Compute the first four partial sums \(S_{1}, \ldots, S_{4}\) for the series having \(n\) th term \(a_{n}\) starting with \(n=1\) as follows. $$ a_{n}=n $$

Short Answer

Expert verified
The first four partial sums are: \( S_1 = 1, S_2 = 3, S_3 = 6, S_4 = 10 \).

Step by step solution

01

Identify the given series

The series provided is a sequence where each term, denoted as \(a_n\), is simply equal to the index \(n\) itself. Thus, the series is: \(a_1 = 1, a_2 = 2, a_3 = 3, a_4 = 4, \ldots\)
02

Calculate the first partial sum \(S_1\)

The first partial sum \(S_1\) is simply the first term of the series. Therefore, \(S_1 = a_1 = 1\).
03

Calculate the second partial sum \(S_2\)

The second partial sum \(S_2\) is the sum of the first two terms of the series. Thus, \(S_2 = a_1 + a_2 = 1 + 2 = 3\).
04

Calculate the third partial sum \(S_3\)

The third partial sum \(S_3\) is the sum of the first three terms of the series. This means \(S_3 = a_1 + a_2 + a_3 = 1 + 2 + 3 = 6\).
05

Calculate the fourth partial sum \(S_4\)

The fourth partial sum \(S_4\) is the sum of the first four terms of the series. Therefore, \(S_4 = a_1 + a_2 + a_3 + a_4 = 1 + 2 + 3 + 4 = 10\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Series
In mathematics, a series is the sum of the terms of a sequence. When you look at a series, imagine a long sequence of numbers, where each individual number, or term, is added to the next. This sums them in a cumulative way. For instance, consider the series formed by adding numbers 1, 2, 3, 4, and so forth. If you write it mathematically, it appears as:
  • \( a_1 + a_2 + a_3 + ... \)
Each of these additions is called a "partial sum," representing how much you get by adding from the first term up to a particular term. The total series is the sum of all these partial sums. With finite series, you add a fixed number of terms. For an infinite series, this sum goes on indefinitely.
Sequence
A sequence is an ordered list of numbers. It's like a list where each number is placed in a specific spot. In sequences, the position of each number is crucial. For example:
  • The sequence \( a_1, a_2, a_3, a_4, \ldots \)
Every sequence has a rule or pattern, which dictates how you determine the next numbers. In our exercise, the sequence where each term is equal to its position starts as 1, 2, 3, 4, etc. This pattern makes it straightforward to find any term when you know its position. It's basically a roadmap for generating numbers, one after another.
Arithmetic Progression
An arithmetic progression is a specific type of sequence where each term after the first is obtained by adding a constant value to the previous term. This constant is known as the "common difference." For example, in the sequence 1, 2, 3, 4, each time you add 1 to get from one term to the next, the common difference is 1:
  • The first term (1)
  • The second term (1 + 1 = 2)
  • The third term (2 + 1 = 3)
  • The fourth term (3 + 1 = 4)
In our example, since each term increases by 1, it forms an arithmetic progression. Arithmetic progressions are simple yet incredibly useful because they allow you to predict any term in the sequence just by knowing the first term and the common difference.
Summation
Summation refers to the operation of adding a series of numbers together. It is denoted by the capital Greek letter sigma (\( \Sigma \)). It provides a compact way to write the sum of a sequence of numbers in mathematics. Consider the summation of the first few terms:
  • \( S_1 = 1 \)
  • \( S_2 = 1 + 2 = 3 \)
  • \( S_3 = 1 + 2 + 3 = 6 \)
  • \( S_4 = 1 + 2 + 3 + 4 = 10 \)
Each of these partial sums can be expressed through summation notation. For instance, summing the first \( n \) natural numbers can be summed up nicely with:\[ S_n = \sum_{i=1}^{n} i \].Summation is powerful because it transforms a long addition process into a compact and understandable form.

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Most popular questions from this chapter

In the following exercises, use either the ratio test or the root test as appropriate to determine whether the series \(\sum_{k=1}^{\infty} a_{k}\) with given terms \(a_{k}\) converges, or state if the test is inconclusive. $$a_{k}=\frac{1 \cdot 4 \cdot 7 \cdots(3 k-2)}{3^{k} k !}$$

In the following exercises, use an appropriate test to determine whether the series converges. $$a_{k}=\left(\frac{k}{k+\ln k}\right)^{2 k}$$

In the following exercises, use an appropriate test to determine whether the series converges. $$\sum_{n=1}^{\infty} \frac{(n+1)}{n^{3}+n^{2}+n+1}$$

In the following exercises, use an appropriate test to determine whether the series converges. $$ a_{n}=1 /\left(\begin{array}{c} n+2 \\ n \end{array}\right) \text { where }\left(\begin{array}{l} n \\ k \end{array}\right)=\frac{n !}{k !(n-k) !} $$

The \(k\) th term of each of the following series has a factor \(x^{k} .\) Find the range of \(x\) for which the ratio test implies that the series converges. $$\sum_{k=1}^{\infty} \frac{x^{2 k}}{3^{k}}$$
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