state whether the series is convergent or divergent
1) 3 + 12 + 48 + ...
2) 4 + 4/5 + 4/25 + ...
#1 surely you don't have to ask about this one -- the terms get ever larger
#2 is simply a geometric series, with |r| < 1, so it converges.
Also, you can tell that the first one is DIVERGENT because the numbers are diverging from one another! In other words, they are getting farther and farther apart
The second one is CONVERGENT because all the numbers are converging to a certain number...in this case they are all getting closer and closer to zero
To determine whether a series is convergent or divergent, we can use several tests, such as the divergence test, geometric series test, integral test, or comparison test. In this case, let's apply the geometric series test.
1) The series 3 + 12 + 48 + ... is defined as a geometric series with a common ratio of 4. To apply the geometric series test, we need to check if the absolute value of the common ratio is less than 1.
|4| = 4 > 1
Since the absolute value of the common ratio is greater than 1, this series diverges.
2) The series 4 + 4/5 + 4/25 + ... can also be recognized as a geometric series with a common ratio of 1/5. Now let's check if the absolute value of the common ratio is less than 1.
|1/5| = 1/5 < 1
Since the absolute value of the common ratio is less than 1, the series converges.
Therefore, the series:
1) 3 + 12 + 48 + ... diverges.
2) 4 + 4/5 + 4/25 + ... converges.