# Limit comparison test

A few people have asked about the limit comparison test during office hours. It’s stated on page 564 of your text, but we can reword it in a way that might make it easier to understand.

First, imagine that instead of assuming that $\lim_{n \to \infty} \frac{a_n}{b_n} = L$, we assumed that $\frac{a_n}{b_n} = L$ for all values of n, not just in the limit.

If $0 < L < \infty$, we could write $a_n = L b_n$ and $b_n = \frac{1}{L}$. Either way, $a_n$ is a multiple of $b_n$ and vice versa. Multiplying a series by a constant (non-zero, non-infinite) doesn’t change whether it converges or diverges. Therefore, either both series $\sum a_n$ and $\sum b_n$ converge or they both diverge….both series have to behave the same way.

If $L$ is very very large, then $b_n = \frac{1}{L} a_n$, that is, $b_n$ is a tiny multiple of $a_n$. Therefore, if you know that $\sum a_n$ converges, then $\sum b_n$ must also converge.

If $L$ is very very small, then $a_n = L b_n$, that is, $a_n$ is a tiny multiple of $b_n$. Therefore, if you know that $\sum b_n$ converges, then $\sum a_n$ must also converge.

None of what I’ve said so far should sound too crazy. What’s interesting about the limit comparison test is that you don’t have to assume that $\frac{a_n}{b_n} = L$ for all values of $n$, you just have to assume that $\frac{a_n}{b_n}$ is close to $L$ only for large values of $n$. If $0 < L < \infty$, what this says is that $a_n$ is eventually almost a multiple of $b_n$ and vice versa. Therefore, either both series $\sum a_n$ and $\sum b_n$ converge or they both diverge…both series have to behave the same way.

Additionally, if you let $L$ actually approach $\infty$, then $b_n$ is eventually almost a tiny multiple of $a_n$. Therefore, if you know that $\sum a_n$ converges, then $\sum b_n$ must also converge.

Finally, if you let $L$ actually approach $0$, then $a_n$ is eventually almost a tiny multiple of $b_n$. Therefore, if you know that $\sum b_n$ converges, then $\sum a_n$ must also converge.

tl;dr: If the individual terms of one series are eventually almost equal to a multiple of the individual terms of another series, then both series have to behave the same way (both converge or both diverge).

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