Let \(4 n^2 + 3 n < a_n < 4 n^2 + 5 n\), \(n = 1, 2, 3, \ldots\). What is the value of \(\operatorname*{lim}\limits_{n \rightarrow \infty} \dfrac{3 a_n}{4 n^2 + 2 n}\) ?

What is the value of \(\displaystyle\sum_{n=1}^\infty \dfrac{2}{n (n + 2)}\) ?

\(\displaystyle\sum_{n=1}^\infty (2 a_n - b_n) = 4\), \(\displaystyle\sum_{n=1}^\infty (a_n + b_n) = 5\). What is the value of \(\displaystyle\sum_{n=1}^\infty (4 a_n - 3 b_n)\) ?

\(\operatorname*{lim}\limits_{n \rightarrow \infty} \dfrac{a_n}{\sqrt{n^2 + 1} - n} = 4\), \(\operatorname*{lim}\limits_{n \rightarrow \infty} (2 n + 1) b_n = 1\). What is the value of \(\operatorname*{lim}\limits_{n \rightarrow \infty} \dfrac{a_n}{b_n}\) ? (Here, \(a_n\), \(b_n\) are sequences.)

What is the value of \(\displaystyle\sum_{n=1}^\infty \dfrac{3 n + 1}{n + 5}\) ?

What is the length of one arch of the cycloid \(x = \theta - \sin \theta\), \(y = 1 - \cos \theta\) ?

Suppose the limaçon is \(r = 2 + \cos \theta\). If the equation of the tangent line at \(t = \dfrac{\pi}{3}\) is \(y = f(x)\), what is the value of \(f(0)\) ?

What quantity of the drug is in the body after the third tablet?

What quantity of the drug remains in the body in the long run?

What is the area of the region that lies inside the circle \(r = 3 \sin \theta\) and outside the cardioid \(r = 1 + \sin \theta\) ?

\(\displaystyle\sum_{n=1}^\infty \dfrac{n!}{2 \cdot 4 \cdot 6 \cdots (2 n)}\)

\(\displaystyle\sum_{n=1}^\infty \left(\dfrac{n + 1}{2 n + 1}\right)^n\)

\(\displaystyle\sum_{n=1}^\infty \dfrac{\cos n \pi}{\sqrt{n}}\)

\(\displaystyle\sum_{n=1}^\infty \dfrac{\ln n}{n^{3 slash 2}}\)

Let the series \(\displaystyle\sum_{n=1}^\infty \dfrac{(3 x + 2)^n}{n^2}\). Find the interval of convergence.

Approximate the function \(f(x) = \sqrt{x}\) by a Taylor polynomial of degree 2 centered at \(a = 4\).

\(\dfrac{1}{1 - x} = \displaystyle\sum_{n=0}^\infty x^n = 1 + x + x^2 + \cdots\), \(|x| < 1\). \(\int \dfrac{1}{1 - x} d x = -\ln(1 - x) + C\). Find a power series representation for \(\ln(1 - x)\) centered at \(x = 0\).

State the Alternating Series Test.