91Ó°ÊÓ

Use the Root Test to determine the convergence or divergence of the series. $$ \sum_{n=1}^{\infty} \frac{(n !)^{n}}{\left(n^{n}\right)^{2}} $$

Write the power series for \((1+x)^{k}\) in terms of binomial coefficients.

In Exercises \(75-88\), determine the convergence or divergence of the series using any appropriate test from this chapter. Identify the test used. \(\sum_{n=1}^{\infty} \frac{(-1)^{n+1} 5}{n}\)

Use the divergence test given in Exercise 71 to show that the series diverges. $$ \sum_{n=1}^{\infty} \frac{n^{3}}{5 n^{4}+3} $$

(a) use Theorem 7.5 to show that the sequence with the given \(n\) th term converges and (b) use a graphing utility to graph the first 10 terms of the sequence and find its limit. \(a_{n}=\frac{1}{3}\left(1-\frac{1}{3^{n}}\right)\)

(a) use Theorem 7.5 to show that the sequence with the given \(n\) th term converges and (b) use a graphing utility to graph the first 10 terms of the sequence and find its limit. \(a_{n}=4+\frac{1}{2^{n}}\)

Determine the convergence or divergence of the series using any appropriate test from this chapter. Identify the test used. \(\sum_{n=1}^{\infty} \frac{5}{n}\)

Prove that \(e\) is irrational. \([\) Hint: Assume that \(e=p / q\) is rational \((p\) and \(q\) are integers) and consider \(\left.e=1+1+\frac{1}{2 !}+\cdots+\frac{1}{n !}+\cdots \cdot\right]\).

Assume that \(|f(x)| \leq 1\) and \(\left|f^{\prime \prime}(x)\right| \leq 1\) for all \(x\) on an interval of length at least \(2 .\) Show that \(\left|f^{\prime}(x)\right| \leq 2\) on the interval.

Determine the convergence or divergence of the series. $$ \frac{1}{200}+\frac{1}{400}+\frac{1}{600}+\frac{1}{800}+\cdots \cdot $$