Determine whether the statement is true or false. Explain your answer.
1. An infinite series converges if its sequence of partial sums is bounded and monotone.
surely
sin(x) is bounded, but not monotone, so it does not converge.
But if it is bounded and monotone, it must approach the bound arbitrarily closely.
Well, if an infinite series converges, that means that the sum of its terms approaches a finite value. So let's take a look at the statement.
First, it says that the sequence of partial sums must be bounded. Well, if the sequence isn't bounded, it's like trying to corral a herd of wild cats - impossible! So, for the series to converge, we need to keep those partial sums under control.
Second, it says that the sequence of partial sums must be monotone. Ah, monotone, like a robot reading a bedtime story in a monotone voice - not very exciting! But in math terms, it means that the sequence of partial sums must be always increasing or always decreasing.
Now, let's think about it. If the sequence of partial sums is bounded and monotone, that means the partial sums are behaving themselves, always getting closer to a specific value and staying within certain boundaries. So it seems logical to say that the infinite series would converge if that were the case.
To sum it all up (pun intended), the statement is true! If the sequence of partial sums is both bounded and monotone, then the infinite series converges.
The statement is true.
An infinite series converges if its sequence of partial sums is both bounded and monotone.
To understand this, let's break it down:
1. Bounded: If the sequence of partial sums of an infinite series is bounded, it means that there exists a fixed value M such that all the partial sums are less than or equal to M. In other words, the sum of all the terms in the series does not "blow up" to infinity. Boundedness is an important criterion for convergence.
2. Monotone: A sequence is said to be monotone if it is either always increasing or always decreasing. In the case of an infinite series, a sequence of partial sums is said to be monotone if it is either always increasing or always decreasing. This means that each partial sum is either greater than or equal to the previous one (for an increasing sequence) or less than or equal to the previous one (for a decreasing sequence).
Now, if an infinite series has both a bounded sequence of partial sums and a monotone sequence of partial sums (either always increasing or always decreasing), then it converges. This means that the sum of the series exists and is a finite value.
On the other hand, if either of these conditions is not met, the series does not converge. If the sequence of partial sums is unbounded or not monotone, then the series diverges, meaning that the sum of the series does not exist or is infinite.
So, in conclusion, if the sequence of partial sums of an infinite series is bounded and monotone, the series converges.
To determine whether the statement is true or false, we can break it down into two separate conditions: boundedness and monotonicity of the sequence of partial sums.
1. Boundedness: We need to examine if the sequence of partial sums is bounded. To do this, we would calculate the partial sum of the series, which is the sum of a finite number of terms. By adding more terms to the series (taking more partial sums), we can observe if there is any pattern of the partial sums tending towards a fixed value. If we find that, after a certain point, the partial sums remain within a certain range or become closer to a constant value, then the series is bounded.
2. Monotonicity: We need to determine whether the sequence of partial sums is monotone. If the sequence is either strictly increasing or strictly decreasing, it is said to be monotone. To check the monotonicity, we can compare the values of two consecutive partial sums and see if they always increase or always decrease. If there is a consistent pattern of the partial sums increasing or decreasing, then the series is said to be monotone.
Now, according to the statement, the infinite series converges if both the sequence of partial sums is bounded and monotone. However, this statement is not entirely true. Both conditions are necessary for the convergence of a series, but they are not sufficient. This means that if a series has a bounded and monotone sequence of partial sums, it may converge, but it is also possible for it to diverge.
To determine the convergence or divergence of a series, we need to consider additional tests such as the ratio test, comparison test, integral test, etc. These tests provide more accurate and reliable information about the convergence or divergence of a series.
In conclusion, the statement that an infinite series converges if its sequence of partial sums is bounded and monotone is false. The boundedness and monotonicity are important characteristics to consider, but they are not enough on their own to determine the convergence of a series. Additional tests are required for a definitive answer.