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Question

  open parentheses lim for n not stretchy rightwards arrow straight infinity of   sum from r equals 1 to n of   open square brackets 1 over 2 to the power of r close square brackets close parentheses to the power of m(Where [ ] denotes GIF) is equal to

  1. 0 for m element ofR

  2. 0 for m space greater than space0

  3. infinity for m less than0

  4. 1 form = 0

The correct answer is:

1 form = 0


    lim for n not stretchy rightwards arrow straight infinity of   open square brackets 1 half plus 1 over 2 squared plus midline horizontal ellipsis plus 1 over 2 to the power of n close square brackets equals lim for n not stretchy rightwards arrow straight infinity of   open square brackets fraction numerator 1 half open parentheses 1 minus 1 over 2 to the power of n close parentheses over denominator 1 minus 1 half end fraction close square brackets
    equals lim for n not stretchy rightwards arrow straight infinity of   open square brackets 1 minus 1 over 2 to the power of n close square brackets equals 0 text  as  end text 1 over 2 to the power of n →∣ 0

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