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

$$\sum_{n \geq 1} \ln \left(\frac{n+1}{n}\right)$$

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

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