What is the sum from k=2 to infinity of 8/(4^{2k)} ?

The sum from k=2 to infinity of 8/(4^{2k)} is 1/30

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sum from k=2 to infinity of 8/(4^{2k)}

k = 2 \sum \infty \frac{8}{4 ^{2 k}}

\frac{1}{30}

Solution steps

k = 2 \sum \infty \frac{8}{4 ^{2 k}}

Apply the constant multiplication rule:

\sum c \cdot a_{n} = c \cdot \sum a_{n}

= 8 \cdot k = 2 \sum \infty \frac{1}{4 ^{2 k}}

4^{2 k} = 1 6^{k}

= 8 \cdot k = 2 \sum \infty \frac{1}{1 6 ^{k}}

n = k \sum \infty (a_{n}) = n = 0 \sum \infty (a_{n}) - a_{0} - a_{1} - \dots a_{k - 1}

= 8 (k = 0 \sum \infty \frac{1}{1 6 ^{k}} - \frac{1}{1 6 ^{0}} - \frac{1}{1 6 ^{1}})

k = 0 \sum \infty \frac{1}{1 6 ^{k}} = \frac{16}{15}

= 8 (\frac{16}{15} - \frac{1}{1 6 ^{0}} - \frac{1}{1 6 ^{1}})

Simplify

8 (\frac{16}{15} - \frac{1}{1 6 ^{0}} - \frac{1}{1 6 ^{1}}) : \frac{1}{30}

= \frac{1}{30}

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What is the sum from k=2 to infinity of 8/(4^{2k)} ?
The sum from k=2 to infinity of 8/(4^{2k)} is 1/30