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Chapter 9: Problem 2

Suppose the sequence \(\left\\{a_{n}\right\\}\) is defined by the explicit formula \(a_{n}=1 / n,\) for \(n=1,2,3, \ldots .\) Write out the first five terms of the sequence.

Short Answer

Expert verified
Question: Write down the first five terms of the sequence defined by the explicit formula \(a_n = \frac{1}{n}\), for \(n=1,2,3, \ldots\). Answer: The first five terms of the sequence are \(1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}\).

Step by step solution

01

Write down the formula for the sequence

The given formula for the sequence is \(a_n = \frac{1}{n}\). We will use this formula to find the first five terms of the sequence by substituting the values of \(n\) from 1 to 5.
02

Calculate the first term (n=1)

To find the first term of the sequence, substitute \(n=1\) into the formula: \(a_1 = \frac{1}{1} = 1\).
03

Calculate the second term (n=2)

To find the second term of the sequence, substitute \(n=2\) into the formula: \(a_2 = \frac{1}{2}\).
04

Calculate the third term (n=3)

To find the third term of the sequence, substitute \(n=3\) into the formula: \(a_3 = \frac{1}{3}\).
05

Calculate the fourth term (n=4)

To find the fourth term of the sequence, substitute \(n=4\) into the formula: \(a_4 = \frac{1}{4}\).
06

Calculate the fifth term (n=5)

To find the fifth term of the sequence, substitute \(n=5\) into the formula: \(a_5 = \frac{1}{5}\).
07

Write down the first five terms of the sequence

Using the calculated values, the first five terms of the sequence \(\{a_n\}\) are: \(1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}\).

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Most popular questions from this chapter

Consider the following infinite series. a. Write out the first four terms of the sequence of partial sums. b. Estimate the limit of \(\left\\{S_{n}\right\\}\) or state that it does not exist. $$\sum_{k=1}^{\infty}(-1)^{k}$$

Given that \(\sum_{k=1}^{\infty} \frac{1}{k^{4}}=\frac{\pi^{4}}{90},\) show that \(\sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k^{4}}=\frac{7 \pi^{4}}{720}.\) (Assume the result of Exercise 63.)

Suppose a ball is thrown upward to a height of \(h_{0}\) meters. Each time the ball bounces, it rebounds to a fraction r of its previous height. Let \(h_{n}\) be the height after the nth bounce and let \(S_{n}\) be the total distance the ball has traveled at the moment of the nth bounce. a. Find the first four terms of the sequence \(\left\\{S_{n}\right\\}\) b. Make a table of 20 terms of the sequence \(\left\\{S_{n}\right\\}\) and determine a plausible value for the limit of \(\left\\{S_{n}\right\\}.\) $$h_{0}=20, r=0.5$$

Consider the following sequences defined by a recurrence relation. Use a calculator, analytical methods, and/or graphing to make a conjecture about the value of the limit or determine that the limit does not exist. $$a_{n+1}=\sqrt{2+a_{n}} ; a_{0}=1, n=0,1,2, \dots$$

Use Exercise 89 to determine how many terms of each series are needed so that the partial sum is within \(10^{-6}\) of the value of the series (that is, to ensure \(R_{n}<10^{-6}\) ). a. \(\sum_{k=0}^{\infty} 0.6^{k}\) b. \(\sum_{k=0}^{\infty} 0.15^{k}\)
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