List the first five terms of the sequence. \

$$\{2 \cdot 4 \cdot 6 \cdots \cdots \cdot(2 n)\}$$

we showed that the harmonic series is divergent. Here we outline another method, making use of the fact that e^x > 1 + x for any x > 0. If sn is the nth partial sum of the harmonic series, show that e^sn > n + 1. Why does this imply that the harmonic series is divergent?

Find a power series representation for the function and determine the interval of convergence. \

$$f(x)=\dfrac{1}{1+9 x^2}$$

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. $\sum_{n=1}^{\infty}(0.6)^{n-1}$

Use the Integral Test to determine whether the series is convergent or divergent. summation 1 to infinity n^2e^-n^3

Determine whether the series is absolutely convergent, conditionally convergent, or divergent. $\sum_{n=1}^{\infty} \frac{(-1)^{n}}{n^{4}}$

Find the sum of the series.

$$1-ln2+(ln2)^2/2!-(ln2)^3/3! + ....$$

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

$$f(x) = x cos x, a = 0$$

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

Find the radius of convergence and interval of convergence of the series. \

$$\sum_{n=1}^n \dfrac{x^n}{n^2}$$

Suppose that ^∞∑n=1 an(an≠0) is known to be a convergent series. Prove that ^∞∑n=1 1/an is a divergent series.

Suppose that f(x) = ^∞∑n=0 cnx^n for all x. If f is an even function, show that c1 = c3 = c5 ... = 0

Find the value of c such that ^∞∑n=0 e^nc = 10

Determine whether the series is absolutely convergent, conditionally convergent, or divergent. \

$$\sum_{n=0}^{\infty} \dfrac{(-3)^n}{n !}$$