This web page contains some examples of the use of the comparison test. The comparison test is perhaps the most basic and the most useful test of all and will be used many times later on, including in the proofs of some important theorems and other tests for convergence such as the ratio test.

I will illustrate both versions of the comparison test here. In the forms I have presented them, the basic form of the comparison test is more general and applies to a greater number of series, and in all cases where the limit comparison test is used the other form may be used instead. The limit comparison test may be conceptually a little harder to state and use, and the final calculation does involves a limit, but generally speaking the algebraic manipulations required tend to be easier. You can, if you prefer, just learn the first version of the test and use that throughout.

Example.

The series ^{2}

**Proof.**

Note that, for all ^{2}
^{2}
^{2}

Example.

The series

**Proof.**

For all

Example.

The series ^{
-}

**Proof.**

Again, as

Example.

Let ^{
}

**Proof.**

In the case ^{
}
^{
}

If 0^{
}
^{
}
^{
}
^{
}

Example.

The series
^{2+3+1
}
^{4+3
2+4+1
}

**Proof.**

Let __ _{
}__=

like

by continuity of +,^{-1}
^{-2}

Now as the limit is 1_{
}_{
}
_{
=
=1
-2
}

Example.

The series
^{2+3+1
}
^{3+3
2+4+1
}

**Proof.**

We perform limit comparision of our series with the harmonic series

by continuity and standard null sequences ^{-1}
^{-2}