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Maths-

General

Easy

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

0 for m $\in$ R
0 for m $>$ 0
$\infty$ for m $<$ 0
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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