5 Sequences Online Quiz Questions

True or false: Every convergent sequence of real numbers is bounded.

Which statement correctly describes the index of a sequence?

Find the limit: $\lim_{n\to\infty}\frac{1}{n^5}$. Enter the numerical value.

Evaluate $\lim_{n\to\infty}\frac{3n^2-1}{n^2+4}$.

True or false: The sequence $a_n=(-1)^n$ converges because it is bounded.

For the sequence $a_n=\frac{2n+1}{n+3}$, enter $\lim_{n\to\infty}a_n$.

Select all conditions that are sufficient, by the Monotone Convergence Theorem, to guarantee that a sequence converges.

Suppose $a_n\to4$. What is $\lim_{n\to\infty}\sqrt{a_n+5}$?

Select all steps that are valid parts of a rigorous strategy for analyzing a recursively defined sequence.

Consider the recursive sequence $a_1=1$ and $a_{n+1}=\sqrt{2+a_n}$. Give a rigorous argument that the sequence converges, and determine its limit.

True or false: Changing the first five terms of a sequence cannot change whether it converges or the value of its finite limit.

What is the correct conclusion about the sequence $a_n=n^2$?

Evaluate $\lim_{n\to\infty}\frac{3n-2}{n+5}$.

Classify the sequence $a_n=1-\frac{1}{n}$ as increasing, decreasing, constant, or neither.

Suppose $a_n\to4$. Find $\lim_{n\to\infty}\sqrt{a_n+5}$.

Use the Squeeze Theorem to determine $\lim_{n\to\infty}\frac{\sin n}{n}$.

What theorem guarantees that an increasing sequence bounded above, or a decreasing sequence bounded below, converges?

The sequence is defined by $a_1=1$ and $a_{n+1}=\sqrt{2+a_n}$. It is known to converge. What is its limit?