The expectation value of the electron's kinetic energy in the hydrogen ground state equals the magnitude of the total energy. What must be the width of a cubic infinite well, in terms of $a_0,$ for its ground state to have this same energy?

Enumerate the six risk factors explained in the lesson.

Find the interval of convergence.

$$\sum(-1)^k \frac{2^k}{3^{k+1}} x^k$$

Determine convergence or divergence by any method.

$$\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } n } { \sqrt { n ^ { 2 } + 1 } }$$

Consider the Fourier sine series of each of the following functions. In this exercise do not compute the coefficients but use the general convergence theorems to discuss the convergence of each of the series in the pointwise, uniform, and $L^2$ senses. (a) $f (x) = x^3$ on (0, l). (b) $f (x) = lx − x^2$ on (0, l). (c) $f (x) = x−2$ on (0, l).

Find the interval of convergence.

$$\sum(-1)^k \frac{k^2}{(k+1) !}(x+3)^k$$

Test the series for convergence.

$$\sum_{k=1}^{\infty} \frac{1}{\sqrt{k^{2}+4}}$$

In this exercise, determine the convergence or divergence of the series.

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

In this task, determine the convergence or divergence of the series.

$$3\displaystyle\sum_{n=1}^{\infty} \frac{1}{n^{0.95}}$$

In this task, determine the convergence or divergence of the series.

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

In this exercise, find the radius of convergence of the power series.

$$\displaystyle\sum_{n=1}^{\infty} \frac{(2 x)^n}{n^2}$$

Determine the convergence or divergence of the series. $\sum_{n=1}^{\infty}\left(\frac{1}{2 n(n+1)}\right)$

Test the given series for convergence.

$$\sum_{k=2}^{\infty} \frac{1}{k\ ln\ k}$$

Determine the convergence or divergence of the integrals.

$$\int_{-1}^{\infty} \frac{\tan ^{-1} x}{(2+x)^3} d x$$