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Limits — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsLimits2 Marks
MCQ
$\lim _{n \rightarrow \infty}\left\{\frac{2}{3}+\left(\frac{2}{3}\right)^2+\left(\frac{2}{3}\right)^3+\ldots+\left(\frac{2}{3}\right)^n\right\}$ is equal to
  • A
    $\frac{1}{2}$
  • ✓
    2
  • C
    1
  • D
    3
✓

Answer

Correct option: B.
2
(B)
$\lim _{n \rightarrow \infty}\left[\frac{2}{3}+\left(\frac{2}{3}\right)^2+\left(\frac{2}{3}\right)^3+\ldots+\left(\frac{2}{3}\right)^n\right]$
$=\lim _{n \rightarrow \infty} \frac{\frac{2}{3}\left(1-\left(\frac{2}{3}\right)^n\right)}{1-\frac{2}{3}}=\lim _{n \rightarrow \infty} 2\left[1-\left(\frac{2}{3}\right)^n\right]$
$=2(1-0)=2^3$

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