Find at least 10 partial sums of the series. Graph both the sequence of terms and the sequence of partial sums on the same screen. Does it appear that the series is convergent or divergent? If it is convergent, find the sum. If it is divergent, explain why. $\sum_{n=1}^{\infty}(0.6)^{n-1}$

Graph the curves

$$y=x^n, 0≤x≤1,$$

for n=0,1,2,3,4... on a common screen. By finding the areas between successive curves, give a geometric demonstration of the fact, that ^∞∑n=1 1/n(n+1) = 1

Determine whether the sequence is convergent or divergent. If it is convergent, find its limit.

$$a_n=\dfrac{\ln n}{\sqrt{n}}$$

Find the Taylor polynomial $T_n(x)$ for the function $f$ at the number $a$. Graph $f$ and $T_n$ on the same screen.
$f(x)=\cos x, \quad a=2 \pi / 3, \quad n=4$

Test the series for convergence or divergence.

$$^∞∑n=1(-1)^n-^1/ln(n+4)$$

The graph of $y=\sqrt{3 x-x^2}$ is given. Use transformations to create a function whose graph is as shown.

Find the sum of the series. summation n=1 to infinite (-1)^n x^42/n!

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If $\sum c_{n} x^{n}$ diverges when x = 6, then it diverges when x = 10.

Find the Maclaurin series for f(x) using the definition of a Maclaurin series. [Assume that f has a power series expansion. Do not show that Rn(x)→0.] Also find the associated radius of convergence. f(x) = ln(1+x)

Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded? an=1/2n+3

Test the series for convergence or divergence. \

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

Find the Taylor polynomial T3(x) for the function f centered at the number a. Graph f and T3 on the same screen.

$$f(x) = e-^x sin x, a = 0$$

Test the series for convergence or divergence. \

$$\sum_{n=1}^{\infty}\left(\dfrac{3 n}{1+8 n}\right)^n$$

Determine whether the series converges or diverges. \

$$\sum_{n=2} \dfrac{1}{n-\sqrt{n}}$$