Loge-1-x1-x-1-x1-x- - a-x2-1-2-x4-3-4-x6-5-6 - - Find a-

Consider, nbsp; log_e[(1+x)^1+x.(1 x)^1 x] =(1+x)log_e(1+x)+(1 x)log_e(1 x) =(1+x) (x x^22+x^33 x^44+x^55 ...∞)+(1 x)( x x^22 x^33 x^44 x^55 ... ...

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Byju's Answer

Standard XII

Mathematics

Summation by Sigma Method

loge[1+x1+x.1...

Question

$l o g_{e} [(1 + x)^{1 + x} . (1 - x)^{1 - x}] = a [\frac{x^{2}}{1.2} + \frac{x^{4}}{3.4} + \frac{x^{6}}{5.6} + . . .]$ . Find $a$ .

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Solution

Consider, ${log}_{e} [(1 + x)^{1 + x} . (1 - x)^{1 - x}]$
$= (1 + x) {log}_{e} (1 + x) + (1 - x) {log}_{e} (1 - x)$
$= (1 + x) (x - \frac{x^{2}}{2} + \frac{x^{3}}{3} - \frac{x^{4}}{4} + \frac{x^{5}}{5} - . . . \infty) + (1 - x) (- x - \frac{x^{2}}{2} - \frac{x^{3}}{3} - \frac{x^{4}}{4} - \frac{x^{5}}{5} - . . . \infty)$
$= 2 (- \frac{x^{2}}{2} - \frac{x^{4}}{4} - \frac{x^{6}}{6} - . . . . .) + 2 (x^{2} + \frac{x^{4}}{3} + . . . . .)$
$= 2 (\frac{x^{2}}{1.2} + \frac{x^{4}}{3.4} + \frac{x^{6}}{5.6} + . . .)$
Comparing with given expression,
$a = 2$

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Similar questions

{log}_{e} [(1 + x)^{1 + x} (1 - x)^{1 - x}] = a (\frac{x^{2}}{1.2} + \frac{x^{4}}{3.4} + \frac{x^{6}}{5.6} + . . . \infty)

Find

a

Q. Find whether the following series are convergent or divergent:

\frac{x}{1.2} + \frac{x^{2}}{3.4} + \frac{x^{3}}{5.6} + \frac{x^{4}}{7.8} + . . .

Q. If

y = \tan^{- 1} \{\frac{\log_{e} (e / x^{2})}{\log_{e} (e x^{2})}\} + \tan^{- 1} (\frac{3 + 2 \log_{e} x}{1 - 6 \log_{e} x})

, then

\frac{d^{2} y}{d x^{2}} =

(a) 2
(b) 1
(c) 0
(d) −1

Q. If the median of the data

x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7}, x_{8}

α

and

x_{1} < x_{2} < x_{3} < x_{4} < x_{5} < x_{6} < x_{7} < x_{8}

, then the median of

x_{3}, x_{4}, x_{5}, x_{6}

If $(1 - x) (1 + x + x^{2} + x^{3} + x^{4})$ is $\frac{31}{32}$ and $x$ is a rational number. Find the value of $1 + x + x^{2} + x^{3} + x^{4} + x^{5}$ .

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loge[(1+x)1+x.(1−x)1−x]=a[x21.2+x43.4+x65.6+...]. Find a.

Consider, loge[(1+x)1+x.(1−x)1−x]=(1+x)loge(1+x)+(1−x)loge(1−x)=(1+x)(x−x22+x33−x44+x55−...∞)+(1−x)(−x−x22−x33−x44−x55−...∞)=2(−x22−x44−x66−.....)+2(x2+x43+.....)=2(x21.2+x43.4+x65.6+...)Comparing with given expression,a=2

$l o g_{e} [(1 + x)^{1 + x} . (1 - x)^{1 - x}] = a [\frac{x^{2}}{1.2} + \frac{x^{4}}{3.4} + \frac{x^{6}}{5.6} + . . .]$ . Find $a$ .