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Chapter 9: Problem 1

Write the initial four terms of the sequence. $$ \left\\{\frac{1}{n}\right\\}_{n=3}^{\infty} $$

Short Answer

Expert verified
The first four terms are \( \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \frac{1}{6} \).

Step by step solution

01

Understand the Sequence Notation

The given sequence is \( \left\{ \frac{1}{n} \right\}_{n=3}^{\infty} \). This means that \( n \) starts from 3 and goes to infinity. We are asked to find the first four terms of this sequence.
02

Calculate the First Term

To find the first term of the sequence, substitute \( n = 3 \) into \( \frac{1}{n} \). This gives \( \frac{1}{3} \). So, the first term is \( \frac{1}{3} \).
03

Calculate the Second Term

For the second term, substitute \( n = 4 \) into \( \frac{1}{n} \). This gives \( \frac{1}{4} \). So, the second term is \( \frac{1}{4} \).
04

Calculate the Third Term

For the third term, substitute \( n = 5 \) into \( \frac{1}{n} \). This results in \( \frac{1}{5} \). Thus, the third term is \( \frac{1}{5} \).
05

Calculate the Fourth Term

For the fourth term, substitute \( n = 6 \) into \( \frac{1}{n} \). This gives \( \frac{1}{6} \). Therefore, the fourth term is \( \frac{1}{6} \).

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Key Concepts

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

Infinite Series
Infinite series refer to sums of an infinite number of terms. They are typically written in the form:
  • a1 + a2 + a3 + ...
In such series, the order and values of the terms follow a specific rule or formula. The sequence provided, \( \{\frac{1}{n}\}_{n=3}^{\infty} \), when summed up, could potentially be analyzed as a series.
A series can converge or diverge:
  • Converge if it approaches a fixed value as the number of terms increases.
  • Diverge if it continually grows larger or oscillates without settling down.
The terms themselves, as in this sequence, do not imply a sum but the foundation of creating an infinite series when summed from the starting term to infinity. Understanding whether the terms converge or not after summation is a key part of deeper series analysis.
Sequence Notation
Sequence notation is used to concisely express the elements of a sequence in calculus. Consider the notation \( \{a_n\} \), which typically indicates a sequence of terms like \( a_1, a_2, a_3, \ldots \). In the sequence \( \{\frac{1}{n}\}_{n=3}^{\infty} \), each term is defined by the expression \( \frac{1}{n} \), starting from \( n=3 \) and potentially extending to infinity.
This helps recognize:
  • The starting point: \( n = 3 \) in this case.
  • The structure: Here, each term diminishes as \( n \) increases.
  • The direction of continuation that extends to infinity.
This form of notation makes it easier to derive specific values in a sequence without listing every individual term.
Term Calculations
Calculating the terms in a sequence involves substituting different values into the form provided. For \( \{\frac{1}{n}\}_{n=3}^{\infty} \), this is how you determine terms:
  • First term: Insert \( n = 3 \), then \( \frac{1}{3} \).
  • Second term: Insert \( n = 4 \), giving \( \frac{1}{4} \).
  • Third term: Insert \( n = 5 \), resulting in \( \frac{1}{5} \).
  • Fourth term: Insert \( n = 6 \), giving \( \frac{1}{6} \).
By understanding this approach, you can efficiently derive the terms of the sequence. This structured process makes it easy to predict or calculate subsequent terms by merely following the pattern prescribed by the sequence formula.

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Most popular questions from this chapter

Determine whether the series converges or diverges. $$ \sum_{n=1}^{\infty}(-1)^{n+1} \frac{1}{n^{1 / 10}} $$

One of Zeno's paradoxes purports to prove that Achilles, who runs 10 times faster than the tortoise, cannot overtake the tortoise, who has a 100 -yard lead. The argument runs as follows: While Achilles runs the 100 yards, the tortoise runs an additional 10 yards. While Achilles runs that 10 yards, the tortoise runs one additional yard. While Achilles runs that yard, the tortoise runs \(\frac{1}{10}\) yard, and so on. Thus Achilles is always behind the tortoise and never catches up. By summing two infinite series, show that Achilles does in fact catch up with the tortoise, and at the same time determine how many yards it takes him to do it.

a. Show that \(\ln \frac{1}{1-x}=\sum_{n=1}^{\infty} \frac{x^{n}}{n}\). b. Using (a), show that \(\ln 2=\sum_{n=1}^{\infty} \frac{1}{n 2^{n}}\). c. Using the fact that $$ \sum_{n=N}^{\infty} \frac{1}{n 2^{n}} \leq \sum_{n=N}^{\infty} \frac{1}{N 2^{n}} $$ estimate \(\ln 2\) with an error less than \(0.01\).

Express the repeating decimal as a fraction. $$ 0.00649649649649 \ldots $$

Express the repeating decimal as a fraction. $$ 0.86400000 \ldots $$
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