Find the series' radius and interval of convergence.

$$\sum_{n=1}^{\infty}\left(1+\frac{1}{n}\right)^n x^n$$

Estimate the error if

$$\cos t ^ { 2 }$$

is approximated by

$$1 - \frac { t ^ { 4 } } { 2 } + \frac { t ^ { 8 } } { 4 ! }$$

in the integral

$$\int _ { 0 } ^ { 1 } \cos t ^ { 2 } d t.$$

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

$$0 . \overline { 23 } = 0.232323 \ldots$$

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

$$\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n + 1 } \frac { 3 + n } { 5 + n }$$

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.)

$$\sum _ { n = 1 } ^ { \infty } \frac { 2 ^ { n } } { 3 ^ { n } }$$

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

$$\frac { 1 } { ( 1 - x ) ^ { 2 } }$$

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

$$\sum _ { n = 1 } ^ { \infty } \frac { 10 n + 1 } { n ( n + 1 ) ( n + 2 ) }$$

Find the value of a for which the limit

$$\lim _ { x \rightarrow 0 } \frac { \sin ( a x ) - \sin x - x } { x ^ { 3 } }$$

is finite and evaluate the limit.

Find the sums of the series.

$$\sum _ { n = 0 } ^ { \infty } e ^ { - n }$$

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

$$\sum _ { n = 1 } ^ { \infty } \frac { 2 + ( - 1 ) ^ { n } } { 1.25 ^ { n } }$$

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

$$\sum_{n=1}^{\infty} \frac{(-1)^n 3^{2 n}(x-2)^n}{3 n}$$

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

$$\sum_{n=1}^{\infty} \frac{(-1)^n 3^{2 n}(x-2)^n}{3 n}$$

Find the series' radius and interval of convergence.

$$\sum_{n=1}^{\infty} \frac{(-1)^n 3^{2 n}(x-2)^n}{3 n}$$

What is a power series? How do you test a power series for convergence? What are the possible outcomes?