document.write( "Question 1195918: This is a calculus question. Can you explain each choice of #4 on https://www.math.purdue.edu/php-scripts/courses/oldexams/serve_file.php?file=16200E3-F2018.pdf ? \n" ); document.write( "
Algebra.Com's Answer #828544 by ikleyn(52781)\"\" \"About 
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document.write( "(I) At small values of real positive quantity x, ln(1+x) is equivalent to x \r\n" );
document.write( "    as a small quantity in Calculus.\r\n" );
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document.write( "    Even more precisely,  0 < \"ln%281%2Bx%29\" <= x\r\n" );
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document.write( "    THEREFORE,  0  <  \"sum%28ln%281%2B1%2Fn%5E2%29%2Cn=1%2Cinfinity%29\" <= \"sum%28%281%2Fn%5E2%29%2Cn=1%2Cinfinity%29\".\r\n" );
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document.write( "    The series  \"sum%28%281%2Fn%5E2%29%2Cn=1%2Cinfinity%29\"  is convergent, which is well known fact in Calculus after Euler. \r\n" );
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document.write( "    So, the series  \"sum%28ln%281%2B1%2Fn%5E2%29%2Cn=1%2Cinfinity%29\"  is convergent, too.\r\n" );
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document.write( "(II) When x is small, sin(x) is equivalent to x as a small quantity in Calculus.\r\n" );
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document.write( "     Therefore, \"sum%28sin%281%2Fn%29%2C+n=1%2Cinfinity%29\"  is equivalent to \"sum%28%281%2Fn%29%2C+n=1%2Cinfinity%29\".\r\n" );
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document.write( "     The series \"sum%28%281%2Fn%29%2C+n=1%2Cinfinity%29\" is harmonic series, and it is divergent - which is\r\n" );
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document.write( "         a well known fact in Calculus.\r\n" );
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document.write( "     It implies that the series \"sum%28sin%281%2Fn%29%2C+n=1%2Cinfinity%29\" is divergent, too.\r\n" );
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document.write( "(III) Estimate  \"sqrt%28n%5E3%29-sqrt%28n%5E3-1%29\" = \"n%5E%283%2F2%29%2A%281-sqrt%281-1%2Fn%5E3%29%29\"  is equivalent to  n^(3/2)*(1/(2n^3)) = (1/2)*(1/n^(3/2)).\r\n" );
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document.write( "      Therefore, it is OBVIOUS that the series  \"sum%28%28sqrt%28n%5E3%29-sqrt%28n%5E3-1%29%29%2Cn=1%2Cinfinity%29\"  is convergent.\r\n" );
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document.write( "Thus we have this conclusion: series (I) converges; series (II) diverges; series (III) converges.\r\n" );
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document.write( "ANSWER.  Option (B).\r\n" );
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\n" ); document.write( "\n" ); document.write( "Comment from student: Thank you. I am confused on the step that you explained  (I took a screenshot of it):
\n" ); document.write( "https://drive.google.com/file/d/1SMF_1rbZAQCY0SzU_9WfjtzSff8WLDbC/view?usp=sharing
\n" ); document.write( "How did you manipulate algebraically?\r
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\n" ); document.write( "\n" ); document.write( "My response:   the question is to explain why \r
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\n" ); document.write( "\n" ); document.write( "         \"1+-+sqrt%281-1%2Fn%5E3%29\"   is equivalent to   \"1%2F%282n%5E3%29\".         (*)\r
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\n" ); document.write( "\n" ); document.write( "Notice that in this problem  n  is a  BIG  value;  so,  \"1%2Fn\"  and  \"1%2Fn%5E3\"  are small quantities.\r
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\n" ); document.write( "\n" ); document.write( "In  Calculus,  if  \" a \"  is a small value,  then \r
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\n" ); document.write( "\n" ); document.write( "         \"sqrt%281%2Ba%29-1\"  is equivalent, in order of smallness,  to  \"a%2F2\".         (**)\r
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\n" ); document.write( "\n" ); document.write( "It is from the area of common knowledge in  Calculus.\r
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\n" ); document.write( "\n" ); document.write( "So,  our estimate  (*)  is a direct consequense of  (**).\r
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\n" ); document.write( "\n" ); document.write( "If you seriously want to understand the solutions of this problem and other similar problems
\n" ); document.write( "from your list,  you should be  FREE  in manipulating basic knowledge of  Calculus,\r
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\n" ); document.write( "\n" ); document.write( "since they are  PRE-REQUIZITES  for solving such problems.\r
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