This web page starts a new section of the course:
infinite series. An infinite series is an expression
such as
This page starts with the definition of convergence for infinite series and three basic but very important results about them. Examples of convergent series will appear in a later page.
An infinite series is an expression of the form
partial sums
,
converges to _{
} as _{
} is called the
Example.
The series _{
=
=1
=12

}.
Thus _{
10=1
} as
Example.
The series _{
=
=1
} is 1 if _{
=12(1)
2
} does not converge
so neither does the series
We shall prove three very important but straightfoward results about
series to get our theory off the ground. The first of these states that,
as with sequences, the convergence of
, and
not on the first few terms—where few
means any finite number
of terms, such as 1000000000. In other words, to show that
Proposition on eventual convergence, or ignoring finitely many terms.
Proof.
This result is similar to the very easy one for sequences
that says: if the subsequence (_{
+
)}
consisting of all terms in (_{
)
} after the
The next result is commonly used in its contrapositive form to show that a series does not converge. It cannot be used to show that a series does converge.
Proof.
Let _{
=
=1
}
be the _{
}
as
Example.
The series
Proof.
The sequence
Our final result here is an application of the monotone convergence theorem to series, and will be used implicitly or explicitly a huge amount in the section on series.
Monotonicity Theorem.
Proof.
There's almost nothing to say, as the argument is presented in the statement of the theorem, except that the final conclusion rests on the monotone convergence theorem for one direction and on the theorem on boundedness for the other direction.
By the proposition on eventual convergence, the monotonicity theorem also applies to any series with only finitely many negative terms: such a series is convergent if and only if the sequence of partial sums is bounded.