document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1185530>Question 1185530</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the Maclaurin series for f(x) using the definition of a Maclaurin series. [Assume that f has a power series expansion. Do not show that <BR>\n" );
document.write( "Rn(x) → 0.]<BR>\n" );
document.write( "f(x) = ln(1 + 2x) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #816676 by </B><A HREF=/tutors/aboutme.mpl?userid=robertb><B>robertb(5830)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=robertb><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=robertb><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=816676\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22f%27%28x%29%22+=+2%2A%281%2B2x%29%5E%28-1%29 ALIGN=MIDDLE  ALT=\"%22f%27%28x%29%22+=+2%2A%281%2B2x%29%5E%28-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22f%27%27%28x%29%22+=+%28-1%29%5E1%2A2%5E2%2A%281%2B2x%29%5E%28-2%29 ALIGN=MIDDLE  ALT=\"%22f%27%27%28x%29%22+=+%28-1%29%5E1%2A2%5E2%2A%281%2B2x%29%5E%28-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22f%27%27%27%28x%29%22+=+%28-1%29%5E2%2A2%5E3%2A2%21%2A%281%2B2x%29%5E%28-3%29 ALIGN=MIDDLE  ALT=\"%22f%27%27%27%28x%29%22+=+%28-1%29%5E2%2A2%5E3%2A2%21%2A%281%2B2x%29%5E%28-3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "....<BR>\n" );
document.write( "Inductively, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%5E%28%28k%29%29%28x%29+=+%28-1%29%5E%28k-1%29%2A2%5Ek%2A%28k-1%29%21%2A%281%2B2x%29%5E%28-k%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >, and so \r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%5E%28%28k%29%29%280%29+=+%28-1%29%5E%28k-1%29%2A2%5Ek%2A%28k-1%29%21 ALIGN=MIDDLE  ALT=\"f%5E%28%28k%29%29%280%29+=+%28-1%29%5E%28k-1%29%2A2%5Ek%2A%28k-1%29%21\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "===> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%281%2B2x%29+=+sum%28+%28f%5E%28%28k%29%29%280%29%2Fk%21%29%2Ax%5Ek%2C+k=0%2Cinfinity%29+=+sum%28+%28%28%28-1%29%5E%28k-1%29%2A2%5Ek%2A%28k-1%29%21%29%2Fk%21%29%2Ax%5Ek%2C+k=1%2C+infinity%29+=+red%28sum%28+%28%28%28-1%29%5E%28k-1%29%2A2%5Ek%29%2Fk%29%2Ax%5Ek%2C+k=1%2C+infinity%29%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );