does the infinite geometric series diverge or converge?
1/7 + 1/14 + 1/28 + 1/56 +
sum∞ = a/(1-r)
= (1/7)/(1-1/2)
= 2/7
so it converges
To determine if an infinite geometric series converges or diverges, we need to examine the common ratio of the series.
In a geometric series, each term is obtained by multiplying the previous term by a constant number called the common ratio (r). In your series, the common ratio can be found by dividing any term by its previous term.
Let's calculate the common ratio in your series:
1/14 divided by 1/7 = (1/14) * (7/1) = 1/2
1/28 divided by 1/14 = (1/28) * (14/1) = 1/2
1/56 divided by 1/28 = (1/56) * (28/1) = 1/2
As you can see, the common ratio (r) in your series is 1/2.
Now, for an infinite geometric series to converge, the absolute value of the common ratio (|r|) must be less than 1. If |r| is greater than or equal to 1, the series will diverge.
In your case, |1/2| = 1/2 which is less than 1.
Therefore, the infinite geometric series 1/7 + 1/14 + 1/28 + 1/56 + ... converges.