document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1137179>Question 1137179</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 1.Find the Maclaurin series for 𝑒 exponential kx, 𝑘 is a real number?\r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #755015 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20850)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=MathLover1><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=MathLover1><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=755015\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Find the Maclaurin series for e^(kx), k is a real number.\r<BR>\n" );
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document.write( "Calculate the derivatives:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f ALIGN=MIDDLE  ALT=\"f\"   DISABLED_style= \"max-width:90%; height:auto;\" >′<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%29=%28e%5E%28kx%29%29 ALIGN=MIDDLE  ALT=\"%28x%29=%28e%5E%28kx%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >′=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ke%5E%28kx%29 ALIGN=MIDDLE  ALT=\"ke%5E%28kx%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f ALIGN=MIDDLE  ALT=\"f\"   DISABLED_style= \"max-width:90%; height:auto;\" >′′<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%29=%28ke%5E%28kx%29%29 ALIGN=MIDDLE  ALT=\"%28x%29=%28ke%5E%28kx%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >′=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k%5E2%2Ae%5E%28kx%29 ALIGN=MIDDLE  ALT=\"k%5E2%2Ae%5E%28kx%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >,…....,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%5E%28n%29%28x%29=k%5En%2Ae%5E%28kx%29 ALIGN=MIDDLE  ALT=\"f%5E%28n%29%28x%29=k%5En%2Ae%5E%28kx%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Then, at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0 ALIGN=MIDDLE  ALT=\"x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > we have\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%280%29=e%5E0=1 ALIGN=MIDDLE  ALT=\"f%280%29=e%5E0=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >,  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f ALIGN=MIDDLE  ALT=\"f\"   DISABLED_style= \"max-width:90%; height:auto;\" >′<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%280%29=ke%5E0=k ALIGN=MIDDLE  ALT=\"%280%29=ke%5E0=k\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f ALIGN=MIDDLE  ALT=\"f\"   DISABLED_style= \"max-width:90%; height:auto;\" >′′<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%280%29=k%5E2%2Ae%5E0=k%5E2 ALIGN=MIDDLE  ALT=\"%280%29=k%5E2%2Ae%5E0=k%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >,…<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%5E%28n%29%280%29=k%5En%2Ae%5E0=k%5En ALIGN=MIDDLE  ALT=\"f%5E%28n%29%280%29=k%5En%2Ae%5E0=k%5En\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Hence, the Maclaurin expansion for the given function is\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5E%28kx%29=sum%28f%5E%28n%29%280%29%28x%5En%2Fn%21%29%2Cn=0%2Cinfinity%29 ALIGN=MIDDLE  ALT=\"e%5E%28kx%29=sum%28f%5E%28n%29%280%29%28x%5En%2Fn%21%29%2Cn=0%2Cinfinity%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Bkx%2B%28k%5E2x%5E2%29%2F2%21%2B%28k%5E3x%5E3%29%2F3%21 ALIGN=MIDDLE  ALT=\"1%2Bkx%2B%28k%5E2x%5E2%29%2F2%21%2B%28k%5E3x%5E3%29%2F3%21\"   DISABLED_style= \"max-width:90%; height:auto;\" >+…=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sum%28%28k%5En%2Ax%5En%29%2Fn%21%2Cn=0%2Cinfinity%29 ALIGN=MIDDLE  ALT=\"sum%28%28k%5En%2Ax%5En%29%2Fn%21%2Cn=0%2Cinfinity%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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