Show how to rearrange the terms of the series from the specified exercise to form a divergent series.

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

Determine the convergence or divergence of the series using any appropriate test. Identify the test used. Σ_(n=1)^∞ n! / n7^n

Determine the convergence or divergence of the series. Identify the test (or tests) you use. There may be more than one correct way to determine convergence or divergence of a given series.

$$\sum_{n=1}^{\infty} n ! e^{-n}$$

Let f be the function given by $f(x)=\frac{2 x}{1+x^{2}}$. Write the first four nonzero terms and the general term of the Taylor series for f about x=0.

Determine the convergence or divergence of the series. Identify the test (or tests) you use. There may be more than one correct way to determine convergence or divergence of a given series.

$$\sum_{n=1}^{\infty} \frac{2^n}{3^n+1}$$

Find a Maclaurin series for the function. xeˉˣ²

Use the series and the function $f(x)$ that it represents from the indicated exercise to find a power series for $\int_0^t f(t) d t$

$$\sum_{n=0}^{\infty}\left(-\frac{1}{2}\right)^n(x-3)^n$$

What is the coefficient of (x - 2)³ in the Taylor series generated by ln x at x = 2?

The approximation e˟ ≈ 1 + x + x²/2 + x³/6 is used on a small intervals about the origin. Estimate the magnitude of the approximation error for │x│ ≤ 0.1.

Find the radius of convergence of the power series. Σ_(n=1)^∞ (4x)^n / n²

determine the radius of convergence of the given power series. ∞∑n=1n!xnnn

Suppose that the radius of convergence of the power series $\Sigma c_n x^n$ is $R$. What is the radius of convergence of the power series $\Sigma c_n x^{2 n}$?

Suppose that the radius of convergence of the power series $\Sigma c_n x^n$ is $R$. What is the radius of convergence of the power series $\Sigma c_n x^{2 n}$?

Show that the series converges absolutely.

$$\sum_{n=0}^{\infty} \frac{(\cos x)^n}{n !+1}$$