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docs/math/real-analysis/3-sequences-series.md
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@@ -16,7 +16,7 @@ $$
(x_n) = (a, ar, ar^2, ar^3, \ldots)
$$
**Example**. The *arithmatic sequence*, given base $a \in \mathbb{R}$ and distance $d \in \mathbb{R}$,
**Example**. The *arithmetic sequence*, given base $a \in \mathbb{R}$ and distance $d \in \mathbb{R}$,
$$
(x_n) = (a, a + d, a + 2d, a + 3d, \ldots)
@@ -52,7 +52,7 @@ $$
2. $x_n \cdot y_n \rightarrow xy$
3. If $x_n \neq 0$ for all $n$, then $\frac{1}{x_n} \rightarrow \frac{1}{x}$
**Theorem**. Suppose $(x_n)$ aand $(y_n)$ are convergent sequences and $N \in \mathbb{N}$. Then,
**Theorem**. Suppose $(x_n)$ and $(y_n)$ are convergent sequences and $N \in \mathbb{N}$. Then,
1. If $x_n \leq y_n$ for all $n \geq N$, then $\lim(x_n) \leq \lim(y_n)$
2. If $x_n \leq a$ for all $n \geq N$, then $\lim(x_n) \leq a$
@@ -110,7 +110,7 @@ $$
## Section 3.7 - Series
**Definition**. Let $(x_n)$ be a sequence in $\mathbb{R}$. Then, the *infinite series genearted by $X$* is the sequence $S = (s_n)$ with terms
**Definition**. Let $(x_n)$ be a sequence in $\mathbb{R}$. Then, the *infinite series generated by $X$* is the sequence $S = (s_n)$ with terms
$$
s_1 = x_1; \; s_{n+1} = s_n + x_{n+1}