Solutions to Problem Set 17

Math 211-03

10-17-2017

[p-series]

1. Determine whether the series $\displaystyle \sum_{n=1}^\infty \dfrac{1}{\sqrt{n}}$ converges or diverges.

$$\sum_{n=1}^\infty \dfrac{1}{\sqrt{n}} = \sum_{n=1}^\infty \dfrac{1}{n^{1/2}}.$$

This is a p-series with $p =
   \dfrac{1}{2} < 1$ . Hence, it diverges.


2. Determine whether the series $\displaystyle \sum_{n=1}^\infty n^{-4}$ converges or diverges.

$$\sum_{n=1}^\infty n^{-4} = \sum_{n=1}^\infty \dfrac{1}{n^4}.$$

This is a p-series with $p = 4 >
   1$ . Hence, it converges.


3. Determine whether the series $\displaystyle \sum_{n=1}^\infty \dfrac{1}{(n + 2)^2}$ converges or diverges.

Let $k = n + 2$ . When $n = 1$ , I have $k = 3$ . So

$$\sum_{n=1}^\infty \dfrac{1}{(n + 2)^2} = \sum_{k=3}^\infty \dfrac{1}{k^2}.$$

This is a p-series with $p = 2 >
   1$ . Hence, it converges.

Note: As a shortcut, I will call series like this "p-series", even though they do not start with index 1.


[Zero Limit Test]

4. Determine whether the series $\displaystyle \sum_{n=3}^\infty \dfrac{5 n + 2}{1 - 8 n}$ converges or diverges. If the series converges, find the exact value of its sum.

$$\lim_{n \to \infty} \dfrac{5 n + 2}{1 - 8 n} = -\dfrac{5}{8} \ne 0.$$

The terms of the series do not go to 0. Hence, the series diverges by the Zero Limit Test.


5. Determine whether the series $\displaystyle \sum_{n=1}^\infty \left(\tan^{-1} n -
   \dfrac{1}{n^2 + 1}\right)$ converges or diverges. If the series converges, find the exact value of its sum.

$$\lim_{n \to \infty} \left(\tan^{-1} n - \dfrac{1}{n^2 + 1}\right) = \dfrac{\pi}{2} - 0 = \dfrac{\pi}{2} \ne 0.$$

The terms of the series do not go to 0. Hence, the series diverges by the Zero Limit Test.


6. Determine whether the series $\displaystyle \sum_{n=1}^\infty \sin (n^2 + 3)$ converges or diverges. If the series converges, find the exact value of its sum.

$$\lim_{n \to \infty} \sin (n^2 + 3) \quad\hbox{is undefined}.$$

The terms of the series do not go to 0. Hence, the series diverges by the Zero Limit Test.


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