When do series converge

The limit of the sequence terms is,. So, as we saw in this example we had to know a fairly obscure formula in order to determine the convergence of this series. In general finding a formula for the general term in the sequence of partial sums is a very difficult process. We will continue with a few more examples however, since this is technically how we determine convergence and the value of a series. This is actually one of the few series in which we are able to determine a formula for the general term in the sequence of partial fractions.

Therefore, the series also diverges. Again, do not worry about knowing this formula. The sequence of partial sums is convergent and so the series will also be convergent. The value of the series is,. As we already noted, do not get excited about determining the general formula for the sequence of partial sums.

Two of the series converged and two diverged. Notice that for the two series that converged the series term itself was zero in the limit. This will always be true for convergent series and leads to the following theorem. Then the partial sums are,. Be careful to not misuse this theorem! This theorem gives us a requirement for convergence but not a guarantee of convergence. In other words, the converse is NOT true. Consider the following two series. The first series diverges. Again, as noted above, all this theorem does is give us a requirement for a series to converge.

Donate Login Sign up Search for courses, skills, and videos. Convergent and divergent sequences. Partial sums: formula for nth term from partial sum. Partial sums: term value from partial sum. Practice: Partial sums intro. Infinite series as limit of partial sums. Next lesson. Current timeTotal duration Google Classroom Facebook Twitter. Video transcript So we've explicitly defined four different sequences here. And what I want you to think about is whether these sequences converge or diverge.

And remember, converge just means, as n gets larger and larger and larger, that the value of our sequence is approaching some value. And diverge means that it's not approaching some value. So let's look at this. And I encourage you to pause this video and try this on your own before I'm about to explain it. So let's look at this first sequence right over here. So the numerator n plus 8 times n plus 1, the denominator n times n minus So one way to think about what's happening as n gets larger and larger is look at the degree of the numerator and the degree of the denominator.

And we care about the degree because we want to see, look, is the numerator growing faster than the denominator? In which case this thing is going to go to infinity and this thing's going to diverge. Or is maybe the denominator growing faster, in which case this might converge to 0? Or maybe they're growing at the same level, and maybe it'll converge to a different number.

So let's multiply out the numerator and the denominator and figure that out. So n times n is n squared. The harmonic series diverges because, even though it increases by smaller and smaller amounts, it will still never actually end at a target, basically for any value n there is some iteration of the harmonic series which has a larger value than that. It just flies away. Maybe the best way to try to answer your question is in terms of theorems. Also the root test and ratio test explain why some series converge or diverge, by comparison to geometric series whose convergence and divergence you can basically take as an axiom when you are talking about why arbitrary series converge or diverge.

This answer is hopefully to increase your intuition about summation. Imagine adding infinitely many non negative, for simplicity numbers together. Roughly speaking, if this addition adds up to a finite number, you say that the series the terms you are summing together converges, and if it doesn't, you say that it diverges.

This is just a natural generalisation of finite addition to infinitely many terms. Lets look at this a little more closer. The result can not be anything finite, can it? But is this enough? The answer is no, as we saw in the example above. To verify that the above example series add up to a finite number or not, you can use some known integral or convergence tests.

In the hope this long-belated musing helps, let's come at the question from the opposite direction. Perhaps it's clear er the answer is "yes". Metaphorically, if you start with a piece of licorice or chocolate-covered spaghetti, or Pocky, which is basically chocolate-covered spaghetti Of course, real confections in our universe are made of finitely many atoms, and cannot be divided into infinitely many pieces of positive size.

You also have to imagine inhabiting a universe where continuum Pocky exists. Your friend Zeno of Elea might eat half of the remaining amount with each bite, or nine-tenths; your friend Salvadori Dali might vary the fraction with each bite. Each scheme of potential consumption gives you a sequence of positive real numbers adding up to a finite total. Sign up to join this community. The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group.

Create a free Team What is Teams? Learn more. Why do some series converge and others diverge? Ask Question. Asked 7 years, 7 months ago. Active 6 years, 6 months ago. Viewed 5k times.

OpieDopee OpieDopee 1, 1 1 gold badge 8 8 silver badges 22 22 bronze badges. This question seems to be extremely broad, though. They behave the way they behave because they behave the way they behave.