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# Geometric Series

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Compute the sum
$$1 + \frac{1}{5} + \frac{2}{5} + \frac{2}{25} + \frac{4}{25} + \frac{4}{125} + \frac{8}{125} + \frac{8}{625} + \dotsb.$$

Apr 15, 2024

#1
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Let's observe the pattern of the terms in the series:

$1, \frac{1}{5}, \frac{2}{5}, \frac{2}{25}, \frac{4}{25}, \frac{4}{125}, \frac{8}{125}, \frac{8}{625}, \dotsb$

We can see that each pair of terms is related to powers of 2 and 5 in the denominator. Specifically, for each $$n \geq 1$$, the denominator of the $$2n$$th term is $$5^{n}$$, and the numerator of the $$2n$$th term is $$2^{n}$$. Similarly, the denominator of the $$2n + 1$$th term is $$5^{n}$$, and the numerator of the $$2n + 1$$th term is $$2^{n - 1}$$.

Let's rewrite the series:

$1, \frac{1}{5}, \frac{2}{5}, \frac{2}{25}, \frac{4}{25}, \frac{4}{125}, \frac{8}{125}, \frac{8}{625}, \dotsb$

$= 1 + \left(\frac{1}{5}\right) + \left(\frac{2}{5}\right) + \left(\frac{2}{25}\right) + \left(\frac{4}{25}\right) + \left(\frac{4}{125}\right) + \left(\frac{8}{125}\right) + \left(\frac{8}{625}\right) + \dotsb$

$= 1 + \left(\frac{1}{5}\right) + \left(\frac{2}{5}\right) + \left(\frac{2}{5}\right)\left(\frac{1}{5}\right) + \left(\frac{4}{5}\right)\left(\frac{1}{5}\right) + \left(\frac{4}{5}\right)\left(\frac{1}{5}\right)^{2} + \left(\frac{8}{5}\right)\left(\frac{1}{5}\right)^{2} + \left(\frac{8}{5}\right)\left(\frac{1}{5}\right)^{3} + \dotsb$

$= 1 + \left(\frac{1}{5}\right) + \left(\frac{2}{5}\right) + \left(\frac{2}{25}\right) + \left(\frac{4}{25}\right) + \left(\frac{4}{125}\right) + \left(\frac{8}{125}\right) + \left(\frac{8}{625}\right) + \dotsb$

Now, we can see that each term is a geometric series. The first term is $$1$$, and the common ratio is $$\frac{1}{5}$$. So, the sum of this series is:

$S_1 = \frac{1}{1 - \frac{1}{5}} = \frac{5}{4}$

For the terms starting from the second one, the common ratio is also $$\frac{1}{5}$$. So, the sum of this series is:

$S_2 = \frac{\frac{1}{5}}{1 - \frac{1}{5}} = \frac{\frac{1}{5}}{\frac{4}{5}} = \frac{1}{4}$

Thus, the total sum of the series is:

$S = S_1 + S_2 = \frac{5}{4} + \frac{1}{4} = \frac{6}{4} = \frac{3}{2}$

Therefore, the sum of the given series is $$\frac{3}{2}$$.

Apr 28, 2024