What makes a geometric series converge

On the other hand, the series with the terms has a sum that also increases with each additional term. However, each time we add in another term, the sum is not going to get that much bigger. This is especially true when we add in terms like. This is only the 21st term of this series, but it's very small. While the ideas of convergence and divergence are a little more involved than this, for now, this working knowledge will do. In fact, we can tell if an infinite geometric series converges based simply on the value of r.

This means it only makes sense to find sums for the convergent series since divergent ones have sums that are infinitely large. This is true even though the formula we gave you technically gives you a number when you put in a 1 and r , even for divergent series.

The other formula is for a finite geometric series , which we use when we only want the sum of a certain number of terms. The n th partial sum of a geometric series is given by:. It simply means that we're only going to add up the first n terms. That might be 5, 10, or 20 terms. The bottom line is we're not adding them all the way to infinity.

This formula is really close to our original formula. The only difference is the 1 — r n. It works for any geometric series regardless of the value of r. Nice try. In other words, we're doubling each term, so our common ratio is 2. That means the series diverges and its sum is infinitely large. Intro Lesson. Lesson: 1a. Lesson: 1b. Lesson: 1c. Lesson: 1d. Lesson: 2a. Lesson: 2b. Lesson: 2c. Intro Learn Practice. Do better in math today Get Started Now. Introduction to sequences 2.

Monotonic and bounded sequences 3. Introduction to infinite series 4. Convergence and divergence of normal infinite series 5. Divergence of harmonic series 8.

The difference of a few terms one way or the other will not change the convergence of a series. In this portion we are going to look at a series that is called a telescoping series. The name in this case comes from what happens with the partial sums and is best shown in an example. By now you should be fairly adept at this since we spent a fair amount of time doing partial fractions back in the Integration Techniques chapter.

If you need a refresher you should go back and review that section. So, what does this do for us? Notice that every term except the first and last term canceled out. This is the origin of the name telescoping series. This also means that we can determine the convergence of this series by taking the limit of the partial sums.

In telescoping series be careful to not assume that successive terms will be the ones that cancel. Consider the following example. The partial sums are,. In this case instead of successive terms canceling a term will cancel with a term that is farther down the list. The end result this time is two initial and two final terms are left. So, this series is convergent because the partial sums form a convergent sequence and its value is,.

Also note that just because you can do partial fractions on a series term does not mean that the series will be a telescoping series. The following series, for example, is not a telescoping series despite the fact that we can partial fraction the series terms. In order for a series to be a telescoping series we must get terms to cancel and all of these terms are positive and so none will cancel. Next, we need to go back and address an issue that was first raised in the previous section.