Find the interval of convergence.

$$\sum \frac{k}{2 k+1} x^{2 k+1}$$

Determine whether the series converges or diverges.

$$\sum \frac{1}{1+2+\cdots+k}$$

The partial sum indicated is used to estimate the sum of the series. Estimate the error.

$$\sum_{k=0}^{\infty}(-1)^k \frac{1}{\sqrt{k+1}} ; \quad s_{80}$$

a. Expand $e^x$ in powers of $x-a$. b. Use the expansion to show that $e^{x_1+x_2}=e^{x_1} e^{x_2}$. c. Expand $e^{-x}$ in powers of $x-a$.

Show that

$$\sum_{k=1}^{\infty} \ln \left(\frac{k+1}{k}\right) \quad \text { diverges }$$

although

$$\ln \left(\frac{k+1}{k}\right) \rightarrow 0 .$$

Estimate to within $0.01$ by using series.

$$\int_0^1 \sin \sqrt{x} d x$$

Find the interval of convergence.

$$\sum \frac{(-1)^k k}{3^{2 k}} x^k$$

Determine whether the series converges or diverges.

$$\sum \frac{k^{k / 2}}{k !}$$

Use Taylor polynomials to estimate the following within $0.01$.

$$\cos 6^{\circ}$$

Determine whether the series converges or diverges.

$$\sum \frac{\ln k}{k^{5 / 4}}$$

Find the interval of convergence.

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

Expand $g(x)$ as indicated. $g(x)=(x-1)^n$ in powers of $x$.

Find the interval of convergence.

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

Estimate to within $0.01$ by using series.

$$\int_0^1 \sin x^2 d x$$