So my question is find a formula for the sum of the first $n$ terms of the series (not sequence):
$$\frac{4}{81}+\frac{4}{729}+\frac{4}{6561}+\frac{4}{59049}+\ldots~\textrm{upto }n\textrm{ terms}$$
How would I do this? I know the formula for writing the sum of a geometric series, but I don't know what to put for r. I know an is 4. Would r be 1/9 or 1/81?
Best Answer
Here, $r$ should be the ratio between successive elements in the sequence. This is computed in your case as 1/9, and you can write arbitrary terms as $4/81 * (1/9)^n$ in general. The finite version of the formula for geometric sums should then provide your answer.