document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1117305>Question 1117305</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Consider the infinite geometric series (linked below). In this image, the lower limit of the summation is \"n=1\".\r<BR>\n" );
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document.write( "As I am not sure how to type this in formatting on this website or how to type this in text this is a link to an image of the formula mentioned above. <BR>\n" );
document.write( "https://postimg.cc/image/hchcwn7jr/\r<BR>\n" );
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document.write( "A) Write the first four terms of the series.\r<BR>\n" );
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document.write( "B) Does the series diverge or converge?\r<BR>\n" );
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document.write( "C) If the series has a sum, find the sum. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #732335 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=732335\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
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document.write( "<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">sum[i=1,infty] [-4*(1/3)^(n-1)]</PRE> is an understandable plain text representation of \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \sum_{i=1}^\infty\,-4\left(\frac{1}{3}\right)^{n-1}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE -4\left(\frac{1}{3}\right)^0\ -4\left(\frac{1}{3}\right)^1\ -4\left(\frac{1}{3}\right)^2\ -4\left(\frac{1}{3}\right)^3 = -4\ -\ \frac{4}{3}\ -\ \frac{4}{9}\ -\ \frac{4}{27}\ \approx\ -5.93\">\r<BR>\n" );
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document.write( "As to convergence/divergence, there are a couple of ways to look at it.\r<BR>\n" );
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document.write( "You can write\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \sum_{i=1}^\infty\,-4\left(\frac{1}{3}\right)^{n-1}\ =\ -4\,\sum_{i=0}^\infty\,\left(\frac{1}{3}\right)^n\">\r<BR>\n" );
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document.write( "But <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \sum_{i=0}^\infty\,\left(\frac{1}{3}\right)^n\"> is a geometric series with <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE r\ =\ \frac{1}{3}\">.  Since any geometric series with <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE -1\ <\ r\ <\ 1\"> converges, this series must converge.\r<BR>\n" );
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document.write( "Alternatively, you can use the Series Ratio Test:  A series where:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \lim_{n\right\infty}\,\frac{a_{n+1}}{a_n}\ =\ L\">\r<BR>\n" );
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document.write( "where <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE L\ <\ 1\"> converges.  Verification of the series ratio test is left as an exercise for the student.\r<BR>\n" );
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document.write( "Either way, the series converges.\r<BR>\n" );
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document.write( "The sum of a geometric series with a common ratio <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large r\"> is given by:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ S\ =\ \frac{1}{1-r}\">\r<BR>\n" );
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document.write( "So\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ -4\,\sum_{i=0}^\infty\,\left(\frac{1}{3}\right)^n\ =\ -4\left(\frac{1}{1\ -\ \frac{1}{3}}\right)\ =\ -6\">\r<BR>\n" );
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document.write( "You can verify the arithmetic for yourself.\r<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi}\ +\ 1\ =\ 0\"><BR>\n" );
document.write( "My calculator said it, I believe it, that settles it<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src=\"http://c0rk.blogs.com/gr0undzer0/darwin-fish.jpg\"><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \ \\r\"> </SPAN>\n" );
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