Bài 105467

Question

Áp dụng đạo hàm tính các tổng sau:
a) ${P_n}(x) = 1 + 2x + 3x^2 + ... + nx^{n - 1}$
b) ${Q_n}(x) = 1^2 + 2^2x + {3^2}{x^2} + ... + {n^2}{x^{n - 1}}$
c) ${S_n}(x) = \frac{1}{2}tan\frac{x}{2} + \frac{1}{4}tan\frac{x}{4} + ... + \frac{1}{{{2^n}}}tan\frac{x}{{{2^n}}}$

hoàng anh thọ · Answer 1 · 5/24/2012 9:19:44 AM

a) ${P_n} = (x + {x^2} + {x^3} + ... + {x^n})' = \left[ {\frac{{x(1 - {x^n})}}{{1 - x}}} \right]$,
    $ = \frac{{1 + n{x^{n + 1}} - (n + 1){x^n}}}{{{{(1 - x)}^2}}}$

b) ${Q_n} = (x.{P_n})' = \left[ {x.\frac{{1 + n{x^{n + 1}} - (n + 1){x^n}}}{{{{(1 - x)}^2}}}} \right]$
    $ = \frac{{1 + x - {{(n + 1)}^2}{x^n} + (2{n^2} + 2n - 1){x^{n + 1}} - {n^2}{x^{n + 2}}}}{{{{(1 - x)}^3}}}$

c) ${S_n} = - \left( {\ln \left| {c{\rm{os}}\frac{x}{2}} \right| + \ln \left| {c{\rm{os}}\frac{x}{4}} \right| + ... + \ln \left| {c{\rm{os}}\frac{x}{{{2^n}}}} \right|} \right)$,
   $ = - \left( {\ln \left| {c{\rm{os}}\frac{x}{2}.c{\rm{os}}\frac{x}{4}c{\rm{os}}\frac{x}{8}...c{\rm{os}}\frac{x}{{{2^n}}}} \right|} \right)$                $(1)$
Chú ý rằng :
$c{\rm{os}}\frac{x}{2}.c{\rm{os}}\frac{x}{4}c{\rm{os}}\frac{x}{8}...c{\rm{os}}\frac{x}{{{2^n}}}$
$ = \frac{1}{{\sin \frac{x}{{{2^n}}}}}.\sin \frac{x}{{{2^n}}}.c{\rm{os}}\frac{x}{{{2^n}}}.c{\rm{os}}\frac{x}{{{2^{n - 1}}}}....c{\rm{os}}\frac{x}{4}.c{\rm{os}}\frac{x}{2}$
$ = \frac{1}{{\sin \frac{x}{{{2^n}}}}}.\frac{1}{2}\sin \frac{x}{{{2^{n - 2}}}}.c{\rm{os}}\frac{x}{{{2^{n - 1}}}}...c{\rm{os}}\frac{x}{4}.c{\rm{os}}\frac{x}{2}$
$ = ... = \frac{1}{{\sin \frac{x}{{{2^n}}}}}.\frac{1}{{{2^n}}}.{\mathop{\rm s}\nolimits} {\rm{inx}}$
Thay vào $(1)$:
${S_n} = - \left( {\ln \left| {\frac{1}{{\sin \frac{x}{{{2^n}}}}}.\frac{1}{{{2^n}}}.{\mathop{\rm s}\nolimits} {\rm{inx}}} \right|} \right)$,
   $\begin{array}{l}
= \left( { - \ln \left| {s{\rm{inx}}} \right| + \ln \left| {\sin \frac{x}{{{2^n}}}} \right| + n\ln 2} \right)\\
= - \cot gx + \frac{1}{{{2^n}}}\cot g\frac{x}{{{2^n}}}.
\end{array}$

Bài 105467

1 Đáp án

Lý thuyết liên quan

Bài 105467

1 Đáp án

Liên quan

Lý thuyết liên quan