Convergence of Series

Posted By on December 28, 2014


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Introduction to Series
The Geometric 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 formuladisplaymath217

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

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2.
If the series tex2html_wrap_inline197 is convergent, then we must havedisplaymath229

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

3.
The geometric seriesdisplaymath231

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

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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 seriesdisplaymath245

is convergent and moreover we have

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Example. Show that the series

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

 

is divergent, even though

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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

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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

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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

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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)

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

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Introduction to Series
The Geometric Series

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Posted by Akash Kurup

Founder and C.E.O, World4Engineers Educationist and Entrepreneur by passion. Orator and blogger by hobby

Website: http://world4engineers.com