Show that the positive terms of the alternating harmonic series form a divergent series. Show the same for the negative terms.

Let $f(x)=(1+x)^{1 / 2}+(1-x)^{1 / 2}$. Find the Maclaurin series for f and use it to find $f^{(4)}(0)$ and $f^{(51)}(0)$.

In given problem, write the first four terms of the sequence $\left\{a_n\right\}$. Then use Theorem D to show that the sequence converges.

$$a_n=\frac{n}{n+1}\left(2-\frac{1}{n^2}\right)$$

Respond with true or false to the following assertion. Be prepared to defend your answer.

Every power series converges for at least two values of the variable.

Find the radius of convergence of

$$where p is a positive integer. \sum_{n=0}^{\infty} \frac{(p n) !}{(n !)^p} x^n$$

In given problem, determine convergence or divergence for each of the series. Indicate the test you use.

$$\sum_{n=2}^{\infty}\left(1-\frac{1}{n}\right)^n$$

For the series given in below problem, determine how large n must be so that using the nth partial sum to approximate the series gives an error of no more than 0.0002.

$$\sum_{k=1}^{\infty} \frac{1}{k(k+1)}$$

Let $y=y(x)=x-\frac{x^3}{3 !}+\frac{x^5}{5 !}-\frac{x^7}{7 !}+\cdots$.

Show that y satisfies the differential equation $y^{\prime \prime}+y=0$ with the conditions y(0)=0 and $y^{\prime}(0)=1$. From this, guess a simple formula for y.

Give an example of two series $\Sigma a_n$ and $\Sigma b_n$, both convergent, such that $\sum a_n b_n$ diverges.

In given problem, indicate whether the given series converges or diverges and give a reason for your conclusion.

$$\sum_{n=1}^{\infty} n^2\left(\frac{2}{3}\right)^n$$

By writing $1 / x=1 /[1-(1-x)]$ and using the known expansion of $1 /(1-x)$, find the Taylor series for $1 / x$ in powers of x-1.

In given problem, find a good bound for the maximum value of the given expression, given that c is in the stated interval. Answers may vary depending on the technique used.

$$\left|\frac{4 c}{\sin c}\right| ;\left[\frac{\pi}{4}, \frac{\pi}{2}\right]$$

Prove that if $\Sigma a_n$ diverges, so does $\Sigma\left|a_n\right|$.

Does the sequence converge or diverge? If the sequence converges, find its limit. $\frac{n}{10}+\frac{10}{n}$