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JEE Main & Advanced Mathematics Sequence & Series Question Bank Self Evaluation Test - Sequences and Series

  • question_answer
    The least value of n (a natural number), for which the sum S of the series \[1+\frac{1}{2}+\frac{1}{{{2}^{2}}}+\frac{1}{{{2}^{3}}}+.....\]differs from \[{{S}_{n}}\] by a quantity \[<{{10}^{-6}}\], is

    A) 21

    B) 20  

    C) 19

    D) None

    Correct Answer: A

    Solution :

    [a]  \[S=\frac{1}{1-\frac{1}{2}}=2\] \[{{S}_{n}}=\frac{1\left( 1-\frac{1}{{{2}^{n}}} \right)}{1-\frac{1}{2}}=2-\frac{1}{{{2}^{n-1}}}\] Given  \[S-{{S}_{n}}<{{10}^{-6}}\] \[\therefore \,\,\,\frac{1}{{{2}^{n-1}}}<{{10}^{-6}}\Rightarrow {{2}^{n-1}}>{{10}^{6}}\] \[\therefore \,\,\,n-1>6{{\log }_{2}}10=\frac{6}{0.3010}n>20\]             \[\left[ \because \,\,\,\frac{6}{.3018}<20 \right]\] \[\therefore \,\,\,n=21\]


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