# 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."

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.