Convergence of Series

Consider the series tex2html_wrap_inline197 and its associated sequence of partial sums tex2html_wrap_inline199 . We will say that tex2html_wrap_inline197 is convergent if and only if the sequence tex2html_wrap_inline199 is convergent. The total sum of the series is the limit of the sequence tex2html_wrap_inline199 , which we will denote by

displaymath207

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 tex2html_wrap_inline199 with the convergence of the sequence of numbers tex2html_wrap_inline211 .

Basic Properties.

1.
Consider the series tex2html_wrap_inline197 and its associated sequence of partial sums tex2html_wrap_inline199 . Then we have the formula

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for any tex2html_wrap_inline219 .
This implies in particular that if we know sequence of partial sums tex2html_wrap_inline199 , one may generate the numbers tex2html_wrap_inline223 since we have

displaymath225

2.
If the series tex2html_wrap_inline197 is convergent, then we must have

displaymath229

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

displaymath231

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

displaymath235

4.
(Algebraic Properties of convergent series) Let tex2html_wrap_inline197 and tex2html_wrap_inline239 be two convergent series. Let tex2html_wrap_inline241 and tex2html_wrap_inline243 be two real numbers. Then the new series

displaymath245

is convergent and moreover we have

displaymath247

Example. Show that the series

\begin{displaymath}\sum_{n \geq 1} \ln \left(\frac{n+1}{n}\right)\end{displaymath}

is divergent, even though

displaymath251

Answer. Note that for any tex2html_wrap_inline219 , we have

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Hence we have (for the associated partial sums)

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Since tex2html_wrap_inline259 , then we have

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

displaymath251

since

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which implies

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

displaymath269

Answer. We have

displaymath271

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

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which gives

displaymath275

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

displaymath277

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

displaymath279

We recognize a geometric series. Since tex2html_wrap_inline281 , then the series is convergent and we have

displaymath283

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Mohamed A. Khamsi
Tue Dec 3 17:39:00 MST 1996

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