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<p><font face="serif">In class today I showed how to answer problem
3 using code -- but because the terms of the series are always
decreasing you could have just as easily answered the question
with some simple logic.</font></p>
<p><font face="serif">For example the terms of the sum to make
<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>π</mi><annotation
encoding="TeX">\pi</annotation></semantics></math> are.</font></p>
<p><font face="serif"><math
xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>4</mn><mfrac><msup><mi>x</mi><mrow><mn>2</mn><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msup><mrow><mn>2</mn><mi>i</mi><mo>−</mo><mn>1</mn></mrow></mfrac></mrow><annotation
encoding="TeX">4 \frac{x^{2i-1}}{2i-1}</annotation></semantics></math><br>
</font></p>
<p><font face="serif">This term oscillates between negative and
positive, but what we want to know is when does its value drop
below 0.001. Remember, in our scenario x=1 so the numerator
will ALWAYS be 1. The problem then reduces to what value of i
reduces the term 4/(2i+1) to 0.001. In this case i=2000.</font></p>
<p><font face="serif">If we do the second part, and look for when
the term drops to less than 1e-10, then i=20,000,000,000 (and
that will take a long time to converge!)<br>
</font></p>
<div class="moz-signature">-- <br>
<b><i>Andrew J. Pounds, Ph.D.</i></b><br>
<i>Professor of Chemistry and Computer Science</i><br>
<i>Director of the Computational Science Program</i><br>
<i>Mercer University, Macon, GA 31207 (478) 301-5627</i></div>
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