First, let's identify what kind of series we're dealing with here.
This series is geometric. A geometric series is made up of terms with a common ratio. In this series, each term is 1/3 times the previous one. For example, the second term (1/15) is 1/3 of the first term (1/5), the third term (1/45) is 1/3 of the second term, and so on.
Here's the general form of a geometric series.
⍺ / (1 - r)
Where:
- ⍺ is the first term of the sequence, and
- r is the common ratio.
The geometric series converges when the absolute value of the ratio, |r|, is less than 1. And it diverges when |r| is greater than or equal to 1.
In this series, the common ratio r = 1/3, so |r| = 1/3. Because 1/3 is less than 1, we would expect the series to converge.
However, when we substitute into the formula ⍺ / (1 - r), we have 1/5 divided by 1 - 1/3, which equals 1/5 divided by 2/3, which equals 3/10.
But when we look at the result closely, the series is actually divergent. Since a geometric series with r less than 1 should always be convergent, there might be something wrong here. Upon further inspection, we see that the series is not a classic geometric series. It has an extra factor in the denominator, a 5 in this case. Because of this additional factor, our series does not fit neatly into the classic form of a geometric series, and our previous convergence test fails.
So, the series: (1)/(5)+(1)/(15)+(1)/(45)+(1)/(81)... is not a classic geometric series and it does not converge; rather, it diverges.