Infinite sum with solution but not sure how it got there

Questions
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1

My summation is (4/3^k±3^k/5^k) from 0 to infinity. I can see the series converges and what the value is 7/2 but no clue how they got there using the ratio test. Can anyone help me trace these steps or understand how to get the steps in wolframalpha or mathmatica?

2

Why not look at each of the two terms separately and then add the two results?