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Chapter 7: Problem 98

Compute the first six terms of the sequence \(\left\\{a_{n}\right\\}=\left\\{\left(1+\frac{1}{n}\right)^{n}\right\\}\) If the sequence converges, find its limit.

Short Answer

Expert verified
The first six terms of the sequence are approximately 2, 2.25, 2.37, 2.44, 2.49 and 2.52. The sequence appears to converge, and its limit is \(e\), Euler's number, approximately 2.71828.

Step by step solution

01

Calculate the First Six Terms

Substitute values of \(n\) from 1 to 6 into the sequence \(\left\{a_{n}\right\}=\left\{\left(1+\frac{1}{n}\right)^{n}\right\}\). \nThe terms will be: \n1. For \(n=1\), \(a_1 = \left(1+\frac{1}{1}\right)^1 = 2\) \n2. For \(n=2\), \(a_2 = \left(1+\frac{1}{2}\right)^2 = 2.25\) \n3. For \(n=3\), \(a_3 = \left(1+\frac{1}{3}\right)^3 \approx 2.37\) \n4. For \(n=4\), \(a_4 = \left(1+\frac{1}{4}\right)^4 \approx 2.44\) \n5. For \(n=5\), \(a_5 = \left(1+\frac{1}{5}\right)^5 \approx 2.49\) \n6. For \(n=6\), \(a_6 = \left(1+\frac{1}{6}\right)^6 \approx 2.52\) Note that the final three results have been rounded off to two decimal places.
02

Analyzing Convergence

Observe the terms of the sequence. The numbers appear to be approaching a certain value as n increases. This suggests that the sequence might be convergent.
03

Determine the Limit

It's known that \(\lim_{n\to \infty} \left(1+\frac{1}{n}\right)^{n} = e \approx 2.71828\) (Euler's number). However, the accuracy depends on the limit of \(n\), for example when \(n->\infty\). As \(n\) increases, the terms of the sequence will approach \(e\), which indicates that the sequence is convergent.

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Most popular questions from this chapter

Suppose that \(\sum a_{n}\) and \(\sum b_{n}\) are series with positive terms. Prove that if \(\lim _{n \rightarrow \infty} \frac{a_{n}}{b_{n}}=0\) and \(\sum b_{n}\) converges, \(\Sigma a_{n}\) also converges.

Prove that if \(\left\\{s_{n}\right\\}\) converges to \(L\) and \(L>0,\) then there exists a number \(N\) such that \(s_{n}>0\) for \(n>N\).

Define a geometric series, state when it converges, and give the formula for the sum of a convergent geometric series.

In Exercises 85 and \(86,\) (a) find the common ratio of the geometric series, \((b)\) write the function that gives the sum of the series, and (c) use a graphing utility to graph the function and the partial sums \(S_{3}\) and \(S_{5} .\) What do you notice? $$ 1+x+x^{2}+x^{3}+\cdots $$

Find the sum of the convergent series. $$ 1+0.1+0.01+0.001+\cdots $$
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