Question

Read the section and make your own summary of the material.

Short answer

1. The Limit of a Convergent Sequence.

2. Basic Limit Rules for Convergent Sequences.

3. More Limit Rules for Convergent Sequences.

4. Sub-sequences of Convergent Sequences Converge.

5. Bounded Monotonic Sequences Always Converge.

6. Convergent Sequences are Bounded

Step-by-step solution

Step 1. Given Information.

The sections given in the book.

Step 2. The Limit of a Convergent Sequence.

Suppose $a_k$is a sequence of real numbers. We say that $\underset{k\to \infty }{lim}{a}_k=L$ for some real number L,or equivalently that $a_k\to L$, if the following statement is true:

For any $\in >0$, there exists some N > 0 such that if k > N, then $a_k\in (L-\in ,L+\in )$.

If $a_k\to L$ for some real number L,then we say that the sequence converges to L. If
no such L exists, then we say that the sequence diverges.

Step 3. Basic Limit Rules for Convergent Sequences

If $\left\{a_k\right\}$and $\left\{b_k\right\}$are convergent sequences with $a_k\to L$and $b_k\to M$as $k\to \infty$, and if c is any constant, then

$$\begin{gathered} \left(i\right)c{a}_k\to cL \\ \left(ii\right)(a_k+b_k)\to M+L \\ \left(iii\right)a_k{b}_k\to LM \\ \left(iv\right)IfM\ne 0,then\frac{a_k}{b_k}\to \frac{L}{M} \end{gathered}$$

Step 4. More Limit Rules for Convergent Sequences.

1. Limits of Functions of Sequences.

2. Uniqueness of Limits for Sequences.

3. Squeeze Theorem for Sequences.

4. The Limit of the Absolute Value of a Sequence.

Step 5. Sub-sequences of Convergent Sequences Converge.

Let $\left\{a_k\right\}$ be a sequence that converges to L. Then every sub-sequence of $\left\{a_k\right\}$ also converges to L.

Step 6. Some other theorems.

1. Convergence and Divergence of Geometric Sequences.

2. Dominance Relationships for Simple Sequences.

3. Convergence of Sequences Based on Dominance Ratios.

4. Bounded Monotonic Sequences Always Converge.

5. Convergent Sequences are Bounded