Infinite series Knowledge-Based Practice

Questions

1. 1.

   Consider the infinite series $\sum_{n=1}^{\infty} \frac{n! \, 2^n}{n^n}$. What is the value of the limit $L = \lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right|$, and what does the Ratio Test conclude about the series?

   • A) $L = 2$, so the series diverges
   • B) $L = 0$, so the series converges
   • C) $L = \frac{e}{2}$, so the series diverges
   • D) $L = \frac{2}{e}$, so the series converges
2. 2.

   What is the sum of the infinite series $\sum_{n=0}^{\infty} \frac{(-1)^n \pi^{2n}}{6^{2n} (2n)!}$?

   • A) $\frac{1}{2}$
   • B) $-\frac{\sqrt{3}}{2}$
   • C) $\frac{\sqrt{2}}{2}$
   • D) $\frac{\sqrt{3}}{2}$
3. 3.

   What is the sum of the infinite series $\sum_{n=1}^{\infty} \frac{2}{n(n+2)}$?

   • A) 1
   • B) $\frac{3}{2}$
   • C) 2
   • D) $\frac{3}{4}$
4. 4.

   What is the interval of convergence of the power series $\sum_{n=1}^{\infty} \frac{(x + 2)^n}{n \cdot 3^n}$?

   • A) $(-5, 1)$
   • B) $[-5, 1)$
   • C) $(-5, 1]$
   • D) $[-5, 1]$
5. 5.

   Which of the following statements correctly describes the behavior of the infinite series $\sum_{n=1}^{\infty} \frac{(-1)^n}{\sqrt{n^2 + 3n}}$?

   • A) It converges absolutely.
   • B) It diverges by the Ratio Test.
   • C) It diverges by the $n$th-term test for divergence.
   • D) It converges conditionally.

Answers and step-by-step solutions appear after you finish the test.