document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=48744>Question 48744</A>This question is from textbook <A HREF=http://www.amazon.com/exec/obidos/ASIN/0534373526/algebrahomeworkh>College Algebra</A><BR>\n" );
document.write( ": <I><SPAN STYLE=\"word-break: break-all;\"> Write the sum without using sigma notation:\r<BR>\n" );
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document.write( "n over (capital sigma)j=1(under) (-1)^j+1 x^j\r<BR>\n" );
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document.write( "Thank you very much! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #32331 by </B><A HREF=/tutors/aboutme.mpl?userid=kev82><B>kev82(151)</B></A><img src=\"/images/stars/stars-09.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=kev82><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=32331\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Hi,\r<BR>\n" );
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document.write( "So we need to find a formula for <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\sum_{j=1}^N\,(-1)^{j+1}x^j\">\r<BR>\n" );
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document.write( "To figure out what's actually happenning, lets write out a few terms (I'm only going to work with even N for now)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?x - x^2 + x^3 - x^4 + x^5 -x^6 \ldots\">\r<BR>\n" );
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document.write( "Or splitting this into two sequences.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?x + x^3 + x^5 + \ldots\"><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?-( x^2 + x ^4 + x^6 + \ldots)\">\r<BR>\n" );
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document.write( "You should be able to see that the first sequence is just a factor of -x different from the first.\r<BR>\n" );
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document.write( "The sequence clearly has N/2 terms(remember I'm assuming N even at moment)\r<BR>\n" );
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document.write( "Using the formula for a geometric series (1-x^n)/(1-x) I'm sure you've seen it, we can see that the sequence we want\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?x^2 + x^4 + x^6 + \ldots + x^{2n}\">\r<BR>\n" );
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document.write( "where <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?n=1+\lfloor N/2 \rfloor\"> is given by\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\frac{ x^{2\lfloor 1+\frac{N}{2} \rfloor} - 1}{x^2 - 1} - 1\">\r<BR>\n" );
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document.write( "Using this and the fact we want <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?1/x\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?-1\"> copies of this sequence we get the general formula (lots of tidying up)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\frac{x^2 - x^{2\left\lfloor 1+\frac{N}{2} \right\rfloor}}{x(x+1)}\">\r<BR>\n" );
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document.write( "However, we haven't dealt with the n=odd case. If you try out a few examples you will note that when n is odd there is an extra term <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?x^N\"> So we need to add this when n is odd. The easiest way to do this is with a function that is 1 for odd and 0 for even.\r<BR>\n" );
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document.write( "Adding <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?-\frac{(-1)^N-1}{2}x^N\"> gives the final answer of\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\frac{x^2 - x^{2\left\lfloor 1+\frac{N}{2} \right\rfloor}}{x(x+1)} -\frac{(-1)^N-1}{2}x^N\">\r<BR>\n" );
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document.write( "It's quite possible I've made a mistake, but I've checked it with a few cases and it works so I'm pretty confident.\r<BR>\n" );
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document.write( "Of course the geometric formula isn't valid for x=1, so neither is this. I'll let you solve that case your self - it's pretty simple though.\r<BR>\n" );
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document.write( "Hope that helps,<BR>\n" );
document.write( "Kev </SPAN>\n" );
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