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Chapter 1: Q14P (page 1)

Question: Use the integral test to find whether the following series converge or diverge. Hint and warning: Do not use lower limits on your integrals.
14. ${鈭�}_{1}^{鈭�} \frac{1}{\sqrt{n^{2} + 9}}$

Short Answer

Expert verified

The series ${鈭�}_{1}^{鈭�} \frac{1}{\sqrt{n^{2} + 9}}$ is divergent.

Step by step solution

01

Definition of convergent and divergent

If the partial sum $S_{n}$ of an infinite series tend to a limit S, then the series is called convergent.

If the partial sum $S_{n}$ of an infinite series do not approach to a limit, the series is called divergent.

The limiting value S is called the sum of the series.

02

Apply the integral test

The given series is ${鈭�}_{1}^{鈭�} \frac{1}{\sqrt{n^{2} + 9}}$ .

Use the integral in the given series as:

${鈭�}_{}^{鈭�} \frac{1}{\sqrt{n^{2} + 9}} 鈥� d n$

Now, solve the integral is as follows:

Use the integral formula $鈭� \frac{1}{\sqrt{x^{2} + a^{2}}} 鈥� d x = \log |x + \sqrt{x^{2} + a^{2}}| + c$ , where c is a constant.

$\begin{array}{rcl} {鈭�}_{}^{鈭�} \frac{1}{\sqrt{n^{2} + 9}} 鈥� d n & = & {[\log |n + \sqrt{n^{2} + 9}|]}^{鈭�} \\ = & [\log |鈭� + \sqrt{鈭� + 9}|] \\ = & 鈭� \end{array}$

Here, the series approaches to infinite, therefore the given series diverges.

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Most popular questions from this chapter

Use the ratio test to show that a binomial series converges for $| x | < 1$

Evaluate the following indeterminate forms by using L鈥橦opital鈥檚 rule and check your results by computer. (Note that Maclaurin series would not be useful here because xdoes not tend to zero, or because a function (In x, for example) is not expandable in a Maclaurin series.

$a) \underset{x 鈫� 蟺}{l i m} \frac{x s i n x}{x - 蟺} b) \underset{x 鈫� 蟺 / 2}{l i m} \frac{l n (2 - s i n x)}{l n (1 + c o s x)} c) \underset{x 鈫� 1}{l i m} \frac{l n (2 - x)}{l n (x - 1)} d) \underset{x 鈫� 鈭�}{l i m} \frac{l n x}{\sqrt{x}} e) \underset{x 鈫� 0}{l i m} x 2 l n x f) \underset{x 鈫� 鈭�}{l i m} x^{n} e^{- x}$

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Show thatis the distance between the points and in the complex plane. Use this result to identify the graphs in Problems $53, 54, 59, a n d 60$ without computation.

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