The infinite series S = 1/5 + 3/5^3 + 5/5^5 + 7/5^7 + 9/5^9 + ... converges. Find S.
To find the sum of the infinite series S, we can use the formula for the sum of an infinite geometric series.
An infinite geometric series has the general form: a + ar + ar^2 + ar^3 + ..., where 'a' is the first term and 'r' is the common ratio.
In this case, the first term 'a' is 1/5, and the common ratio 'r' is (3/5)^2 = 9/25.
The sum of an infinite geometric series S can be calculated using the formula: S = a / (1 - r), provided that |r| < 1.
Plugging in the values, we have:
S = (1/5) / (1 - 9/25)
= (1/5) / (16/25)
= (1/5) * (25/16)
= 25/80
= 5/16
Therefore, the sum of the infinite series S is 5/16.