A $3.0-\mathrm{M} \Omega$ resistor is connected in series with an initially uncharged $0.28-\mu \mathrm{F}$ capacitor. This arrangement is then connected across four 1.5-V batteries (also in series). (a) What is the maximum current in the circuit and when does it occur? (b) What percentage of the maximum current is in the circuit after 4.0 s? (c) What is the maximum charge on the capacitor and when does it occur? (d) What percentage of the maximum charge is on the capacitor after 4.0 s? (e) How much energy is stored in the capacitor after one time constant has elapsed?

What can you say about such a series?

(a). Find the first four nonzero terms of the Taylor series for the given function centered at $a$. (b). Write the power series using summation notation.

$$f(x)=\sin x, a=\pi / 2$$

Use Theorem $20$ to find the series' interval of convergence and, within this interval, the sum of the series as a function of $x$.

$$\sum_{n=0}^{\infty} \frac{(x-1)^{2 n}}{4^n}$$

Verify that the infinite series diverges. ∑_(n=1)^∞ n² / (n²+1)

If the radius of convergence of the power series $\sum_{n-0}^{\infty} c_n x^n$ is $10$, what is the radius of convergence of the series $\Sigma_{n-1}^{\infty} n c_n x^{n-1}$ ? Why?

Use power series operations to find the Taylor series at $x=0$ for the following function.

$$\dfrac{5x^2}{2}-5+5 \cos x.$$

Expand $f(x)=|x|-x,-1<x<1$, in a Fourier series.

Decide if the statement is true or false. Assume that the Taylor series for a function converges to that function. Give an explanation for your answer.

The Taylor series for $f$ converges everywhere $f$ is defined.

Find the Maclaurin series for the function.

$$f(x) = e^6x$$

(a). Find the first four nonzero terms of the Taylor series for the given function centered at $a$. (b). Write the power series using summation notation.

$$f(x)=e^x, a=\ln 2$$

Find the Maclaurin series for the function. x⁹ - 2x² + 1.

Determine whether each series is arithmetic or geometric. Then evaluate the finite series for the specified number of terms. $1+0.01-0.98-\ldots ; n=5$

Find the Maclaurin series for the function. f(x) = e^x²/2