# 微积分网课代修|微分学代写Differential calculus代考|MATH350 Can we add infinities? Subtract? Divide? Multiply?

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## 微积分网课代修|微分学代写Differential calculus代考|Can we add infinities? Subtract? Divide? Multiply?

We have demonstrated that in our computations of limits we can replace any sequence with its limit and continue doing the algebra. This conclusion doesn’t apply to divergent sequences!

Sequences that approach infinity diverge, technically, but they provide useful information about the pattern exhibited by the sequences. Such a sequence can also be used to create a convergent sequence:
$$a_{n}=m \text { and } b_{n}=\frac{1}{n} .$$

## 微积分网课代修|微分学代写Differential calculus代考|Limits of Rational Functions at Infinity

Every sequence is a function; it just happens to have a special kind of domain. Then, why do they deserve a special attention?

Example 1.6.1: recursive limits
Recursive definitions are very common; even the simplest banking requires one to use them:

If you say that you will contribute $\$ 2000$every year, you are stating that the next year’s balance will be$\$2000$ higher than the last:
$$a_{n+1}=a_{n}+2000 .$$

If you say that your bank will pay $5 \%$ per year, you are stating that the next year’s balance will be $1.05$ times higher than the last:
$$b_{n+1}=b_{n} \cdot 1.05 \text {. }$$
Of course, we then derive the $n$ th-term formulas for these sequences:

repeated deposits starting from 0 :
$$a_{n}=2000 \cdot n$$

compounded interest starting from $\$ 2000$: $$b_{n}=2000 \cdot 1.05^{n} .$$ Then it is clear from these formulas that the limits are infinite. However, what if we carry out both of the strategies? The recursive formula is still very clear: $$c_{n+1}=\left(c_{n}+2000\right) \cdot 1.05 .$$ But there is no$n$th-term formula! How do we even prove that the limit is infinite? Indirectly, by comparing$c_{n}$with either$a_{n}$or$b_{n}$. ## 微积分网课代修|微分学代写Differential calculus代考|Can we add infinities? Subtract? Divide? Multiply? 我们已经证明，在我们的极限计算中，我们可以用它的极限替换任何序列并继续进行代 数。这个结论不适用于发散序列! 从技术上讲，接近无穷大的序列存在分歧，但它们提供了有关序列所展示模式的有用信 息。这样的序列也可用于创建收敛序列: $$a_{n}=m \text { and } b_{n}=\frac{1}{n} .$$ ## 微积分网课代修|微分学代写Differential calculus代考|Limits of Rational Functions at Infinity 每个序列都是一个函数；它恰好有一种特殊的域。那么，为什么它们值得特别关注呢? 示例 1.6.1：递归限制 递归定义非常普遍；即使是最简单的银行业务也需要使用它们: 如果你说你会贡献$\$2000$ 每年，你都说下一年的余额将是 $\$ 2000$高于上一个: $$a_{n+1}=a_{n}+2000 .$$ 如果你说你的银行会支付$5 \%$每年，你说下一年的余额将是$1.05$比上次高几倍: $$b_{n+1}=b_{n} \cdot 1.05 .$$ 当然，我们接着推导出$n$这些序列的 th 项公式: 从 0 开始的重复存款: $$a_{n}=2000 \cdot n$$ 复利从$\$2000$ :
$$b_{n}=2000 \cdot 1.05^{n} .$$

$$c_{n+1}=\left(c_{n}+2000\right) \cdot 1.05 .$$