The_Z_Factor said:
I am extremely confused right now, concerning the limits and I guess the limit rules... I'm either reading it wrong or its just not explaining it well enough for me to understand. Basically, I am lost on this entire chapter of limits. This is kinda what I think is going on.
There are limits to finite series and a finite series is a series of numbers that continue up until a stopping point or number, like 1,2,3,4...10. Then there are infinite series, which never end, but converge to a number, which I guess would be infinite, so in my mind, these partial sums, which are always getting smaller(?), because they are fractions(?), will never reach this point, and become an infinitesimal, which is a number so small that Thompson says it can be 'thrown out'.
Are you clear on the distinction between "sequences" and "series"? A sequence is a list of numbers: 1, 2, 3, 4, ... 10 is a finite
sequence of numbers. If an infinite sequence "converges to a number" it would
not be infinite. "converging" always means "getting close and closer to a finite number:. The infinite list of numbers, 1/2, 2/3, 3/4, 4/5, ..., n/(n+1) converges to 1. Notice that the numerator is always less than the denominator so every number in the sequence is less than 1. If you divide both numerator and denominator by n, you get 1/(1+ 1/n). As n gets larger and larger that fraction 1/n gets smaller and smaller (the limit of the sequence 1, 1/2, 1/3, ..., 1/n is 0).
But then you talk about "partial sums". A "series" is a
sum of numbers. You can always add a finite sequence, but most infinite sequences cannot be added- the "series" does not converge. The "partial sums" of an infinite series are the finite sums. for the series
\Sum_{n=1}^\infty \frac{n}{n+1}
The partial sums would be 1/2, 1/2+ 2/3= 7/6, 1/2+ 2/3+ 3/4= 23/12, etc. Those numbers are clearly getting larger and larger and the series does NOT "converge". It is possible to show that if a sequence does not converge
to 0 then the corresponding series (sum) does not converge.
\Sum_{n=1}^\infty \frac{1}{n}[/itex]<br />
has the property that, even though the individual terms go to 0, the series does <b>not</b> converge. <br />
\Sum_{n=1}^\infty \frac{1}{2^n}[/itex]<br />
on the other hand, <b>does</b> converge. It converges to 1.<br />
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I must say that I dislike talking about &quot;infinitesmals&quot; in a stituation like this. It is possible to do it rigorously but that requires some deep results from logic.<br />
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However, it says in this book that these numbers can go <b>beyond</b> this &#039;limit&#039;, somehow, in an infinite series. My first question is how is this possible?
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</blockquote> I&#039;m afraid I don&#039;t know how to answer that because I don&#039;t know what &#039;limit&#039; you are talking about or what you mean by going &quot;beyond&quot; a limit.<br />
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My understanding of this whole limit of infinite series is (and is probably wrong, because I&#039;m totally confused), is that let&#039;s say I have a square. Corner A of this square to corner B of this square is 10 yards. This 10 yards is the limit, of an infinite number of squares in a row, perhaps. So first I go, 5 yards, then 2.5 yards, etc. I will never reach this particular SQUARE&#039;S limit..but it doesn&#039;t or isn&#039;t relevant to the infinite number of squares because we are solving whatever it is we&#039;re trying to solve for this one particular &#039;limit&#039; square.
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</blockquote> You are confusing things slightly by talking about &quot;squares&quot;. You really are talking about a line segment. Yes, if you go 5 yards along the 10 yard line segment, then 2.5 yards, then 1.25, etc., after any finite number of steps you have not gone the entire distance but <b>in the limit</b> you will reach a total of 10 yards. That is, in fact, precisely the example I gave before, multiplied by 10:<br />
\Sum_{n=1}^\infty \frac{10}{2^n}= 10(1)= 10[/itex]&lt;br /&gt;
Both this example and mine are &amp;quot;geometric series&amp;quot;.&lt;br /&gt;
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I can&amp;#039;t make heads or tails out of &amp;quot;but it doesn&amp;#039;t or isn&amp;#039;t relevant to the infinite number of squares because we are solving whatever it is we&amp;#039;re trying to solve for this one particular &amp;#039;limit&amp;#039; square.&amp;quot;. What do you mean by &amp;quot;solving whatever it is we&amp;#039;re trying to solve&amp;quot;?&lt;br /&gt;
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So is that at least partly right?
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&lt;/blockquote&gt; A little bit to unclear to be either right or wrong!&lt;br /&gt;
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Lastly, the book gave me some examples of what the Gardner calls, &amp;quot;the integral limit of any repeating decimal.&amp;quot; &lt;br /&gt;
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So my last question is what is the point of solving this integral limit of any decimal?
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&lt;/blockquote&gt; &amp;quot;the integral limit of any repeating decimal.&amp;quot; Could you post some of these examples? The only limit I can connect with a repeating decimal is the infinite series given by its digits: .333...= .3+ .03+ .003+ .0003+ ... That&amp;#039;s also a &amp;quot;geometric series and it&amp;#039;s easy to show that that converges to 1/3. Indeed, the definition of &amp;quot;decimal representation of a number&amp;quot; is that it is the sum of the infinite series formed by its digits.
