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Chapter 10: Problem 4

Each of Exercises \(1-6\) gives a formula for the \(n\) th term \(a_{n}\) of a sequence \(\left\\{a_{n}\right\\} .\) Find the values of \(a_{1}, a_{2}, a_{3},\) and \(a_{4} .\) $$ a_{n}=2+(-1)^{n} $$

Short Answer

Expert verified
The values are \(a_1=1\), \(a_2=3\), \(a_3=1\), \(a_4=3\).

Step by step solution

01

Understand the Given Formula

The given formula for the sequence is \( a_n = 2 + (-1)^n \). This formula involves two components: a constant term (2) and an alternating term \((-1)^n\) which dictates the sign change based on whether \(n\) is odd or even.
02

Calculate \(a_1\)

Substitute \(n=1\) into the formula: \(a_1 = 2 + (-1)^1 = 2 - 1 = 1\). Thus, the first term of the sequence is \(a_1 = 1\).
03

Calculate \(a_2\)

Substitute \(n=2\) into the formula: \(a_2 = 2 + (-1)^2 = 2 + 1 = 3\). Therefore, the second term of the sequence is \(a_2 = 3\).
04

Calculate \(a_3\)

Substitute \(n=3\) into the formula: \(a_3 = 2 + (-1)^3 = 2 - 1 = 1\). Hence, the third term of the sequence is \(a_3 = 1\).
05

Calculate \(a_4\)

Substitute \(n=4\) into the formula: \(a_4 = 2 + (-1)^4 = 2 + 1 = 3\). As a result, the fourth term of the sequence is \(a_4 = 3\).

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Key Concepts

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

Alternating Sequences
An alternating sequence is a type of sequence in which the signs switch between positive and negative as you move from one term to the next. This characteristic creates a pattern of changing values that can make sequences particularly interesting to study. Alternating sequences often involve terms like
  • \((-1)^n\) which results in alternating signs between positive and negative depending on whether \(n\) is even or odd.
When \(n\) is even, \((-1)^n\) equals a positive one, while if \(n\) is odd, \((-1)^n\) equals negative one. This pattern influences the position and direction of each term, creating an alternating pattern. Recognizing the alternating nature of a sequence is crucial for predicting future terms and understanding the sequence's overall behavior.
Term Calculation
Calculating terms in a sequence involves substituting a specific value of \(n\) into the given sequence formula to find the corresponding term. For the sequence given by \(a_n = 2 + (-1)^n\), calculating each term requires a few steps:
  • Identify which term of the sequence you need (e.g., \(a_1\), \(a_2\), etc.).
  • Substitute this term's position as \(n\) into the formula.
  • Simplify the expression to find the value of the term.
For example, to calculate \(a_2\): substitute \(n = 2\) into the formula, leading to\[a_2 = 2 + (-1)^2 = 2 + 1 = 3\]By applying this method systematically, any term in an alternating sequence can be easily determined.
Sequence Formula
The sequence formula is the backbone of understanding how a sequence behaves. In mathematics, this formula provides a rule for finding each term of the sequence based on its position (\(n\)). For example, in the sequence formula \(a_n = 2 + (-1)^n\):
  • The formula defines each term \(a_n\) of the sequence based on \(n\).
  • It's composed of a constant term, which here is 2, providing a baseline value.
  • It includes an alternating term \((-1)^n\) that changes the sign of 1, depending on whether \(n\) is even or odd.
Understanding the components and arrangement of a sequence formula is essential for predicting the sequence and solving related problems. Always keep in mind how each element affects the result when \(n\) is substituted in.
Sign Change in Sequences
Sign change in sequences, particularly alternating sequences, plays a fundamental role in their structure and properties. This change occurs due to terms like \((-1)^n\), reflecting mathematical rules governed by the parity of \(n\):
  • If \(n\) is even, the term \((-1)^n\) maintains a positive sign, adding to the constant in the formula.
  • If \(n\) is odd, \((-1)^n\) results in a negative one, therefore subtracting from the constant in the formula.
These alternating signs create a zigzag pattern in the sequence's graph and numerical list. By acknowledging the sign change, one can quickly determine how a sequence behaves, ensuring accurate predictions and calculations for each term.

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Most popular questions from this chapter

Proof of Theorem 21 Assume that \(a=0\) in Theorem 21 and that \(f(x)=\sum_{n=0}^{\infty} c_{n} x^{n}\) converges for \(- R < x < R .\) Let \(g(x)=\sum_{n=1}^{\infty} n c_{n} x^{n-1} .\) This exercise will prove that \(f^{\prime}(x)=g(x)\) that is, \(\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}=g(x)\) $$ \begin{array}{c}{\text { a. Use the Ratio Test to show that } g(x) \text { converges for }} \\ {-R < x < R .} \\ {\text { b. Use the Mean Value Theorem to show that }} \\ {\frac{(x+h)^{n}-x^{n}}{h}=n c_{n}^{n-1}} \\ {\text { for some } c_{n} \text { between } x \text { and } x+h \text { for } n=1,2,3, \ldots}\\\\{\text { c. Show that }} \\ {\quad g(x)-\frac{f(x+h)-f(x)}{h}|=| \sum_{n=2}^{\infty} n a_{n}\left(x^{n-1}-c_{n}^{n-1}\right) |} \\ {\text { d. Use the Mean Value Theorem to show that }} \\\ {\frac{x^{n-1}-c_{n}^{n-1}}{x-c_{n}}=(n-1) d_{n-1}^{n-2}} \\ {\text { for some } d_{n-1} \text { between } x \text { and } c_{n} \text { for } n=2,3,4, \ldots}\\\\{\text { e. Explain why }\left|x-c_{n}\right| < h \text { and why }} \\ {\left|d_{n-1}\right| \leq \alpha=\max \\{|x|,|x+h|\\}} \\ {\text { f. Show that }} \\ {\left|g(x)-\frac{f(x+h)-f(x)}{h}\right| \leq|h| \sum_{n=2}^{\infty}\left|n(n-1) a_{n} \alpha^{n-2}\right|}\\\\{\text { g. Show that } \sum_{n=2}^{\infty} n(n-1) \alpha^{n-2} \text { converges for }- R < x < R \text { . }} \\ {\text { h. Let } h \rightarrow 0 \text { in part (f) to conclude that }} \\ {\quad \lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}=g(x)}\end{array} $$

Estimate the value of \(\sum_{n=2}^{\infty}\left(1 /\left(n^{2}+4\right)\right)\) to within 0.1 of its exact value.

Is it true that if \(\sum_{n=1}^{\infty} a_{n}\) is a divergent series of positive numbers, then there is also a divergent series \(\sum_{n=1}^{\infty} b_{n}\) of positive numbers with \(b_{n}

Find the first three nonzero terms of the Maclaurin series for each function and the values of \(x\) for which the series converges absolutely. \(f(x)=\frac{x^{3}}{1+2 x}\)

Use a CAS to perform the following steps for the sequences. a. Calculate and then plot the first 25 terms of the sequence. Does the sequence appear to be bounded from above or below? Does it appear to converge or diverge? If it does converge, what is the limit \(L\) ? b. If the sequence converges, find an integer \(N\) such that \(\quad\left|a_{n}-L\right| \leq 0.01\) for \(n \geq N .\) How far in the sequence do you have to get for the terms to lie within 0.0001 of \(L ?\) $$ a_{n}=\frac{8^{n}}{n !} $$
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