# 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