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{SECT 0 {EXCHG {PARA 200 "" 0 "" {TEXT 204 11 "Two Series:" }{TEXT 
204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 199 "(1) Prove the following tw
o series (formed by f(n) and g(n)) converge analytically. (2) Compare \+
the speed of convergence of these two series. (3) Prove that both thes
e two series converge to  ln(2). " }{TEXT 204 0 "" }}{PARA 200 "" 0 ""
 {TEXT 204 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 205 44 "f:=pr
oc(n) sum(((-1)^(k-1))*1/k,k=1..n) end;" }{MPLTEXT 1 205 0 "" }}}
{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 205 29 "Sum(((-1)^(k-1))*1/k,k=1
..5);" }{MPLTEXT 1 205 0 "" }}{PARA 201 "> " 0 "" {MPLTEXT 1 205 0 "" 
}}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 205 37 "g:=proc(n) sum(1/(k*2
^k),k=1..n) end;" }{MPLTEXT 1 205 0 "" }}{PARA 201 "> " 0 "" {MPLTEXT 
1 205 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 205 22 "Sum(1/(k*2
^k),k=1..5);" }{MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 201 "> " 0 "" 
{MPLTEXT 1 205 51 "f(1);f(2);f(3);f(4);f(5);f(6);f(7);f(8);f(9);f(10);
" }{MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 205 0 
"" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 205 50 "plot(f(x),x=1..10,
y=0..1,style=point,color=black);" }{MPLTEXT 1 205 0 "" }{MPLTEXT 1 
205 1 "\n" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 205 30 "plot(f(x),
x=1..10,style=line);" }{MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 201 "> " 0 
"" {MPLTEXT 1 205 22 "taylor(ln(x+1),x=0,5);" }{MPLTEXT 1 205 0 "" }}}
{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 205 20 "taylor(ln(x),x=1,5);" }
{MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 205 0 "" 
}}}{EXCHG {PARA 200 "" 0 "" {TEXT 204 101 "When substituting  x =1 int
o the expression above, we see the alternating series converges to ln \+
(2)." }{TEXT 204 0 "" }}}{PARA 202 "" 0 "" {TEXT -1 0 "" }}}{MARK "0 0
 0" 0 }{VIEWOPTS 1 1 0 15 10 1804 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }