Solutions of the Examples in Higher Algebra
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1 (x^ X* x^ \ + ^(,3-4+^--) = log(l + a;)-^{x-log(l + x)} + i |-a; + |-%log(l + a:)l /, 2 IV, „ > 3 1 = (^ + S + ^)l°S(^ + *)-2-^- XXIX.] SUMMATION OF SERIES. 169 21. (e'^ + 1)" = c"* + Cie**^"* + e2c'"-2>"^ + CjeC'-'"' + . . .
{e" - 1)"= e"»; - Cjc("-«^ + cje"'-^^ - Cje'"-"* + . . .
.-. 2{Cie'»-«»+C3e»-3W + C5c(''-«"=+...} = (e"'+l)n-(e»:-l)".
Equate ooefBoients of x^; then, if S denote the required series, 2(S we have —= the coefficient of s^in (e^ + l)"- (e'-l)"; 11 that i§, in f^ + x+'^X or... in 2^-^n(x + ^\ + "^"~^^ 2''-=x2; .-. S=n.2»-2 + n(m-l)2»-'=n(ji + l)2"-».
22. (1) When » ia odd, - 2" _ 1 / 1 1 \ _ "»-(2'.-l){2"+i+.l) ~3 U»-l''"2»+i + iy ' and when n is even, «„= ^2„^i)^3„+i_i) = J (gS^ + 2^+^ : and 3S,„^,= (1 + 1) - Q + ^) + Q + 1) - (^ + ^) + ...
+ \^22»>+i-i "^ 2^+2:ri j ; 1 that is, 352ot=1- 2n+i + {-l)»+i' (2) The general term of the series 7, 17, 31, 49, 71,... is 7 + 10(7i-l) + 2(n-l)(«-2) or 2n^ + in-l; 2»i° + 4n + l ^ 1 /"l , _?_ 1 y n i 1\ /I 2 IN /I 2 IN .•.2S„=(j + 2 + 3J-(2"'5+i) + (3 + 4 + 5)-- = i+^+(-1)'-^^i+M)"-\T2- 170 SUMMATION OF SEMES.
What to read after Solutions of the Examples in Higher Algebra?
You can find similar books in the "Read Also" column, or choose other free books by H S Henry Sinclair Hall to read onlineMoreLess
To quickly assess the difficulty of the text, read a short excerpt:
1 (x^ X* x^ \ + ^(,3-4+^--) = log(l + a;)-^{x-log(l + x)} + i |-a; + |-%log(l + a:)l /, 2 IV, „ > 3 1 = (^ + S + ^)l°S(^ + *)-2-^- XXIX.] SUMMATION OF SERIES. 169 21. (e'^ + 1)" = c"* + Cie**^"* + e2c'"-2>"^ + CjeC'-'"' + . . .
{e" - 1)"= e"»; - Cjc("-«^ + cje"'-^^ - Cje'"-"* + . . .
.-. 2{Cie'»-«»+C3e»-3W + C5c(''-«"=+...} = (e"'+l)n-(e»:-l)".
Equate ooefBoients of x^; then, if S denote the required series, 2(S we have —= the coefficient of s^in (e^ + l)"- (e'-l)"; 11 that i§, in f^ + x+'^X or... in 2^-^n(x + ^\ + "^"~^^ 2''-=x2; .-. S=n.2»-2 + n(m-l)2»-'=n(ji + l)2"-».
22. (1) When » ia odd, - 2" _ 1 / 1 1 \ _ "»-(2'.-l){2"+i+.l) ~3 U»-l''"2»+i + iy ' and when n is even, «„= ^2„^i)^3„+i_i) = J (gS^ + 2^+^ : and 3S,„^,= (1 + 1) - Q + ^) + Q + 1) - (^ + ^) + ...
+ \^22»>+i-i "^ 2^+2:ri j ; 1 that is, 352ot=1- 2n+i + {-l)»+i' (2) The general term of the series 7, 17, 31, 49, 71,... is 7 + 10(7i-l) + 2(n-l)(«-2) or 2n^ + in-l; 2»i° + 4n + l ^ 1 /"l , _?_ 1 y n i 1\ /I 2 IN /I 2 IN .•.2S„=(j + 2 + 3J-(2"'5+i) + (3 + 4 + 5)-- = i+^+(-1)'-^^i+M)"-\T2- 170 SUMMATION OF SEMES.
What to read after Solutions of the Examples in Higher Algebra?
You can find similar books in the "Read Also" column, or choose other free books by H S Henry Sinclair Hall to read onlineMoreLess
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