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Chapter 9: Problem 76

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, 2.52. The sequence converges to the value 'e' (approximately 2.718).

Step by step solution

01

Calculate the first six terms

To find the first six terms of the sequence, substitute \(n\) for integers from 1 to 6 into the sequence definition \(a_{n} = (1+1/n)^n\).\n- For \(n=1\), \(a_1 = (1+1/1)^1=2\)\n- For \(n=2\), \(a_2 = (1+1/2)^2=2.25\)\n- For \(n=3\), \(a_3 = (1+1/3)^3\approx2.37\)\n- For \(n=4\), \(a_4 = (1+1/4)^4\approx2.44\)\n- For \(n=5\), \(a_5 = (1+1/5)^5\approx2.49\)\n- For \(n=6\), \(a_6 = (1+1/6)^6\approx2.52\)
02

Determining if the sequence converges

This sequence is well known in calculus and it converges to the number \(e\) which is approximately 2.718. The limit is recognizable to those familiar with the definition of the number 'e'. The general rule is that as \(n\) gets larger, the value of \(a_n = (1+1/n)^n\) gets closer and closer to \(e\). So we can write that as a limit, \(\lim_{{n \to \infty}} (1+1/n)^n = e\)
03

Conclusion

So now we know that the first six terms of the sequence are approximately 2, 2.25, 2.37, 2.44, 2.49, 2.52 and the series converges to \(e\) as \(n\) approaches infinity.

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Most popular questions from this chapter

Use the Ratio Test or the Root Test to determine the convergence or divergence of the series. \(1+\frac{2}{3}+\frac{3}{3^{2}}+\frac{4}{3^{3}}+\frac{5}{3^{4}}+\frac{6}{3^{5}}+\cdots\)

Finding a Limit In Exercises \(59-62,\) use the series representation of the function \(f\) to find \(\lim _{x \rightarrow 0} f(x)\) (if it exists). $$ f(x)=\frac{\sin x}{x} $$

Find the values of \(x\) for which the series converges. $$\sum_{n=0}^{\infty} 3(x-4)^{n}$$

Using a Binomial Series In Exercises \(17-26,\) use the binomial series to find the Maclaurin series for the function. $$ f(x)=\frac{1}{(1+x)^{2}} $$

Finding a Taylor Polynomial Using Technology In Exercises \(75-78\) , use a computer algebra system to find the fifth-degree Taylor polynomial, centered at \(c\) , for the function. Graph the function and the polynomial. Use the graph to determine the largest interval on which the polynomial is a reasonable approximation of the function. $$ f(x)=\sin \frac{x}{2} \ln (1+x), \quad c=0 $$
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