sum_{k=1}^{infty} sum_{r=0}^k frac{1}{3^k} bi | vedclass

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(D) We know that the sum of binomial coefficients is $\sum_{r=0}^k \binom{k}{r} = 2^k$. Substituting this into the given expression: $\sum_{k=1}^{\inf

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$2$

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(D) We know that the sum of binomial coefficients is $\sum_{r=0}^k \binom{k}{r} = 2^k$. Substituting this into the given expression: $\sum_{k=1}^{\infty} \frac{1}{3^k} \sum_{r=0}^k \binom{k}{r} = \sum_{k=1}^{\infty} \frac{1}{3^k} (2^k)$ $= \sum_{k=1}^{\infty} \left(\frac{2}{3}\right)^k$ This is an infinite geometric series with first term $a = 2/3$ and common ratio $r = 2/3$. The sum of an infinite geometric series is given by $S = \frac{a}{1-r}$. $S = \frac{2/3}{1 - 2/3} = \frac{2/3}{1/3} = 2$.

$\sum_{k=1}^{\infty} \sum_{r=0}^k \frac{1}{3^k} \binom{k}{r}$ is equal to

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