document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1198622>Question 1198622</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Is the following series,  5 -10/3 +20/9 -40/27.<BR>\n" );
document.write( "Convergent, or divergent?One of the following is the correct answer. Which one?\r<BR>\n" );
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document.write( "A) divergent <BR>\n" );
document.write( "B) convergent  </SPAN>\n" );
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document.write( "Answer: <font color=red>Convergent</font>\r<BR>\n" );
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document.write( "Reason:\r<BR>\n" );
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document.write( "The jump from 5 to -10/3 is \"times -2/3\"<BR>\n" );
document.write( "The jump from -10/3 to 20/9 is \"times -2/3\"<BR>\n" );
document.write( "The jump from 20/9 to -40/27 is \"times -2/3\"\r<BR>\n" );
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document.write( "We see that this sequence is geometric with common ratio r = -2/3\r<BR>\n" );
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document.write( "Because -1 < r < 1, this particular infinite geometric series is convergent.\r<BR>\n" );
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document.write( "Extra info:<BR>\n" );
document.write( "To compute the value of 5 - 10/3 + 20/9 - 40/27 + ...<BR>\n" );
document.write( "we'll use this formula<BR>\n" );
document.write( "S = a/(1-r)<BR>\n" );
document.write( "where in this case, a = 5 is the first term.<BR>\n" );
document.write( "So,<BR>\n" );
document.write( "S = a/(1-r)<BR>\n" );
document.write( "S = 5/(1-(-2/3))<BR>\n" );
document.write( "S = 3<BR>\n" );
document.write( "The infinitely many terms add up to 3. <BR>\n" );
document.write( "Realistically we can only add up finitely many terms, which means the sum will steadily approach 3 but never actually arrive at this exact value.<BR>\n" );
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