Find the sum of the following series to infinity : 1/3 + 1/5^2 + 1/3^2 + 1/5^4 + 1/3^5 + 1/5^6 + ......∞

Question

Find the sum of the following series to infinity :

\(\frac{1}{3} + \frac{1}{5^2} + \frac{1}{3^2} + \frac{1}{5^4} + \frac{1}{3^5}+ \frac{1}{5^6}+ .....\infty\)

Answer 1

We observe that above progression possess a common ratio, but alternatively , adjacent terms are not possessing a common ratio. So, it consists of 2 geometric progressions.

Let us denote the two progressions with S1 and S2 

∴ S = S₁ + S₂

Sum of infinite GP = \(\frac{a}{1-r}\) ,where a is the first term and r is the common ratio. 

Note: We can only use the above formula if |r|<1 

Clearly, a = \(\frac{1}{3}\) and r = 1/9

Sum of infinite GP = \(\frac{a}{1-r}\),where a is the first term and r is the common ratio. 

Note: We can only use the above formula if |r|<1

Clearly, a = \(\frac{1}{25}\) and r = 1/25