Does
3/4 - 3/5 + 3/6 - 3/7 + 3/8....
Converge or Diverge?
To determine whether the series
3/4 - 3/5 + 3/6 - 3/7 + 3/8....
converges or diverges, we need to examine the behavior of the terms in the series as we continue to add them up.
First, let's observe the pattern of the series: The numerator remains constant at 3, while the denominator increases by 1 at each term.
To simplify the analysis, we can rewrite the series as:
(3/4) - (3/5) + (3/6) - (3/7) + (3/8)....
Now, let's consider the behavior of the terms as we move along the series. We notice that:
- The denominator increases indefinitely.
- The numerator remains constant.
Since the denominator of each term in the series increases indefinitely, while the numerator is constant, the terms themselves converge to zero as the series progresses.
Ultimately, the convergence or divergence of the series is determined by the sum of these terms. In other words, if the sum of the terms converges to a finite value, the series converges; if the sum diverges (increases without bound), the series diverges.
To determine whether the series converges or diverges, it is necessary to evaluate the sum of the terms. One way to do this is to apply the concept of the alternating series test.
In this series, the terms alternate between positive and negative values. The alternating series test states that if an alternating series satisfies two conditions - the terms decrease in absolute value as we move along the series, and the terms converge to zero - then the series converges.
In our case, the terms of the series decrease in absolute value (each term has a smaller absolute value than its preceding term), and the terms converge to zero (as we observed earlier).
Thus, based on the alternating series test, we can conclude that the given series converges.