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8. sequence and series — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHS8. sequence and series1 Mark
MCQ
The sum of the series $\frac{1}{x+1}+\frac{2}{x^{2}+1}+\frac{2^{2}}{x^{4}+1}+\ldots . .+\frac{2^{100}}{x^{2^{100}}+1}$ when $x=2$ is :
  • A
    $1+\frac{2^{101}}{4^{101}-1}$
  • B
    $1+\frac{2^{100}}{4^{101}-1}$
  • C
    $1-\frac{2^{100}}{4^{100}-1}$
  • ✓
    $1-\frac{2^{101}}{4^{101}-1}$
✓

Answer

Correct option: D.
$1-\frac{2^{101}}{4^{101}-1}$
d
$\mathrm{S}=\frac{1}{\mathrm{x}+1}+\frac{2}{\mathrm{x}^{2}+1}+\frac{2^{2}}{\mathrm{x}^{4}+1}+\ldots \frac{2^{100}}{\mathrm{x}^{2^{100}}+1}$

$\mathrm{~S}+\frac{1}{1-\mathrm{x}}=\frac{1}{1-\mathrm{x}}+\frac{1}{\mathrm{x}+1}+\ldots=\frac{2}{1-\mathrm{x}^{2}}+\frac{2}{1+\mathrm{x}^{2}}+\ldots$

$\mathrm{S}+\frac{1}{1-\mathrm{x}}=\frac{2^{101}}{1-\mathrm{x}^{2^{101}}}$

Put $\mathrm{x}=2$

$\mathrm{S}=1-\frac{2^{101}}{2^{2^{101}}-1}$

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