Find the series' radius and interval of convergence.

$$\sum_{n=1}^{\infty} n^n x^n$$

Express each of the numbers as the ratio of two integers.

$$0 . \overline { 7 } = 0.7777 . .$$

The test, which we state without proof, is an extension of the Ratio Test.

Raabe's Test: If $\sum_{n=1}^{\infty} u_n$ is a series of positive constants and there exist constants $C, K$, and $N$ such that

$$\frac{u_n}{u_{n+1}}=1+\frac{C}{n}+\frac{f(n)}{n^2},$$

where $|f(n)|<K$ for $n \geq N$, then $\sum_{n=1}^x u_n$ converges if $C>1$ and diverges if $C \leq 1$.

Show that the results of Raabe's Test agree with what you know about the series $\sum_{n=1}^{\infty}\left(1 / n^2\right)$ and $\sum_{n=1}^{\infty}(1 / n)$.

Which of the series converge, and which diverge? Give reasons for your answers. (When you check an answer, remember that there may be more than one way to determine the series’ convergence or divergence.)

$$\sum _ { n = 0 } ^ { \infty } \frac { - 2 } { n + 1 }$$

Which of the series converge, and which diverge? Use any method, and give reasons for your answers.

$$\sum _ { n = 1 } ^ { \infty } \left( \frac { n } { 3 n + 1 } \right) ^ { n }$$

Use power series operations to find the Taylor series at x = 0 for the function.

$$x \tan ^ { - 1 } x ^ { 2 }$$

Which of the series converge absolutely, which converge, and which diverge? Give reasons for your answers.

$$\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n + 1 } \frac { 1 + n } { n ^ { 2 } }$$

Which of the series converge absolutely, which converge conditionally, and which diverge? Give reasons for your answers.

$$\sum _ { n = 1 } ^ { \infty } \frac { 1 } { \sqrt { n } }$$

Find a polynomial that will approximate F(x) throughout the given interval with an error of magnitude less than

$$10^{-3}.$$

$$F ( x ) = \int _ { 0 } ^ { x } \sin t ^ { 2 } d t , \quad [ 0,1 ]$$

Use any method to determine if the series converges or diverges. Give reasons for your answer.

$$\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n } \left( 1 - \frac { 3 } { n } \right) ^ { n }$$

For what values of $x$ does the series converge $(c)$ conditionally?

$$\sum_{n=1}^{\infty}(\ln n) x^n$$

For what values of $x$ does the series converge (b) absolutely,

$$\sum_{n=1}^{\infty}(\ln n) x^n$$

Find the series' radius and interval of convergence.

$$\sum_{n=1}^{\infty}(\ln n) x^n$$

Which of the series converge, and which diverge? Use any method, and give reasons for your answers.

$$\sum _ { n = 3 } ^ { \infty } \frac { 5 n ^ { 3 } - 3 n } { n ^ { 2 } ( n - 2 ) \left( n ^ { 2 } + 5 \right) }$$