## Convergence of Series

Consider the series and its associated
sequence of partial sums . We will say that is **convergent** if and only if the sequence
is convergent. The total sum of the series is the limit of
the sequence , which we will denote by

So as you see the convergence of a series is related to the
convergence of a sequence. Many do some serious mistakes in confusing
the convergence of the sequence of partial sums with the
convergence of the sequence of numbers .

**Basic Properties.**

**1.**
- Consider the series and its associated
sequence of partial sums . Then we have the formula

for any .

This implies in particular that if we know sequence of partial sums
, one may generate the numbers since we have

**2.**
- If the series is convergent,
then we must have

In particular, if the sequence we are trying to add does not converge
to 0, then the associated series is divergent.

**3.**
- The geometric series

converges if and only if |*q*|<1. Moreover we have

**4.**
- (Algebraic Properties of convergent series) Let
and be
two convergent series. Let and be two real numbers.
Then the new series

is convergent and moreover we have

**Example.** Show that the series

is divergent, even though

**Answer.** Note that for any , we have

Hence we have (for the associated partial sums)

Since ,
then we have

which implies that the series is divergent. Indeed, we do have

since

which implies

**Example.** Check that the following series is convergent and find its
total sum

**Answer.** We have

Using the above properties, we see here that we are dealing with two
geometric series which are convergent. Hence the original series is
convergent and we have

which gives

**Example.** Check that the following series is convergent and find its
total sum

**Answer.** First we need to clean the expression (by using algebraic manipulations)

We recognize a geometric series. Since , then the series is convergent and we have

**
**

**
[Trigonometry]
[Calculus]
**** **
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[Algebra]
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*Mohamed A. Khamsi *

Tue Dec 3 17:39:00 MST 1996

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