DE7 said:Hey all, it's been a while since I've done series and I have a quick question. How would I show the convergence or divergence of [itex]\sum \left(\sqrt{n+1}-\sqrt{n}\right)[/itex]? The ratio test is inconclusive I think, and I'm not sure how I would go about doing the root test. Or is there a series I could compare it to for the comparison test? Anyways, thanks, just need a little refresher.
Series convergence refers to the behavior of an infinite series, where the sum of all its terms approaches a finite value as the number of terms increases. If the sum of the terms approaches a specific value, the series is said to converge. If the sum does not approach a specific value, the series is said to diverge.
There are various tests that can be used to determine if a series converges or diverges, such as the ratio test, the root test, and the integral test. These tests involve analyzing the behavior of the terms in the series and comparing them to known convergent or divergent series.
One quick method for determining if a series converges is by using the comparison test. This involves comparing the series to a known convergent or divergent series with similar terms. If the known series converges, then the series in question also converges. If the known series diverges, then the series in question also diverges.
No, a series can only converge to one value. This is because the definition of convergence states that the sum of the terms approaches a specific value, meaning that the series can only approach and ultimately equal one value.
The rate of convergence refers to how quickly the terms in a series approach the finite value that the series converges to. A series with a faster rate of convergence will reach its finite value in fewer terms, while a series with a slower rate of convergence will take longer to reach its finite value. This can impact the speed and accuracy of calculations involving the series.