Limit Rules for Sequences

Suppose two (infinite) sequences $a_1, a_2, a_3, \ldots$ and $b_1, b_2, b_3, \ldots$ have limits $a$ and $b$ respectively. In other words, $\lim_{n\to\infty}a_n=a$ and $\lim_{n\to\infty}b_n=b$.

The linearity of limits says the limit of the sequence $a_1+b_1, a_2+b_2, a_3+b_3,\ldots$ is $a+b$.

We can summarize this as the following formula: $\lim_{n\to\infty}a_n + \lim_{n\to\infty}b_n = \lim_{n\to\infty}(a_n+b_n)$.

We should use this formula with caution. The formula holds true only if we know the limits of both sequences $a_1, a_2, a_3, \ldots$ and $b_1, b_2, b_3, \ldots$ exist. However, this is the only restriction of the usage.

Now if you are given a sequence $s_1, s_2, s_3, \ldots$ and you can express each term $c_n$ as the sum of $a_n$ and $b_n$, the first thing you might want to try is the linearity of limits! Make sure that the limits of $a_1, a_2, a_3, \ldots$ and $b_1, b_2, b_3, \ldots$ exist.

Furthermore, we can even have the linearity of limits for three sequences. Suppose three (infinite) sequences $a_1, a_2, a_3, \ldots$, $b_1, b_2, b_3, \ldots$ and $c_1, c_2, c_3, \ldots$ have limits $a$, $b$ and $c$ respectively. Then $$\lim_{n\to\infty}(a_n+b_n+c_n)=a+b+c.$$

You probably would have guessed that this is true for four sequences, five sequences and more. And yes! This is exactly the case as long as all the sequences converge.

For the most part, the limit rules for functions carry over to sequences without a hitch.