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in Geometric Progressions by (15.7k points)

Find the sum of each of the following infinite series :

\(\frac{2}{5} + \frac{3}{5^2}+ \frac{2}{5^3} + \frac{3}{5^4} +..... \infty\)

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1 Answer

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This geometric series is the sum of two geometric series:

Sum of geometric series:

\(\frac{2}{5} + \frac{2}{5^3}+ \frac{2}{5^5}\) + ...... 

Here, a = \(\frac{2}{5}\) 

Sum of geometric  series : 

\(\frac{3}{5^2} + \frac{3}{5^4} + \frac{4}{5^6} + ...... \infty\)

Here, a = \(\frac{3}{5^2}\)

∴Sum of the given infinite series = sum of both the series \(\frac{5}{12} + \frac{1}{8} = \frac{(5\times 2) + (1 \times 3)}{24}\)

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