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- How to solve an infinite series where each term is itself an infinite series?

I had a random thought about infinite series the other day while watching a math video. Let's say we have an infinite series where each term in the series is itself another infinite series. How would one go about finding the sum?

For example, let's say we have the series ##a_1+a_2+a_3...## where ##a_n = (\frac{1}{2})^n+(\frac{1}{4})^n+(\frac{1}{8})^n...##

I assume the series converges since each term is the reciprocal powers of two series raised to a power (without the leading 1/1 term, which would make it diverge), which should fall off quite quickly. However it's been a while since I did any math work involving series so I'm a bit unsure. Thoughts? Is solving something like this fundamentally any different from solving a 'plain' series?

For example, let's say we have the series ##a_1+a_2+a_3...## where ##a_n = (\frac{1}{2})^n+(\frac{1}{4})^n+(\frac{1}{8})^n...##

I assume the series converges since each term is the reciprocal powers of two series raised to a power (without the leading 1/1 term, which would make it diverge), which should fall off quite quickly. However it's been a while since I did any math work involving series so I'm a bit unsure. Thoughts? Is solving something like this fundamentally any different from solving a 'plain' series?