# Comparison tests for series

## Main Question or Discussion Point

Comparison Test

Consider the two infinite series $\sum a_n$ and $\sum b_n$ with $a_n,b_n \geq 0$ and $a_n \leq b_n$ for all $n$.
If $\sum b_n$ is convergent so is $\sum a_n$.
If $\sum a_n$ is divergent so is $\sum b_n$.

Limit Comparison Test

Consider the two infinite series $\sum a_n$ and $\sum b_n$ with $a_n \geq 0$ and $b_n > 0$ for all $n$. Define:
$$c = \lim_{n \rightarrow \infty} \frac{a_n}{b_n}$$
If $c$ is positive and finite then either both series converge or both series diverge.

Does it matter if the two series we're comparing have different starting points? I don't think it matters because we can always strip out a finite number of terms from one of the series until both series have the same starting point.

http://tutorial.math.lamar.edu/Classes/CalcII/SeriesCompTest.aspx

But according to this link, "in order to apply this test we need both series to start at the same place."

## Answers and Replies

pwsnafu
Does it matter if the two series we're comparing have different starting points? I don't think it matters because we can always strip out a finite number of terms from one of the series until both series have the same starting point.
You are removing a finite number of terms because you care about different starting points.

Ssnow
Gold Member
Sometimes it possible to change index in order to have series from a determinate point, as example from $\sum_{i=1}^{\infty}a_{i}$ putting $i-1=k$ we have $\sum_{k=0}^{\infty}a_{k+1}$...

Mark44
Mentor
Does it matter if the two series we're comparing have different starting points? I don't think it matters because we can always strip out a finite number of terms from one of the series until both series have the same starting point.
The limit is as $n \to \infty$, so the starting points of the two series don't matter.