# Finding the Interval of Convergence for a Series

• roam
In summary: It means that for every two terms there is a one in between that cancels it out. This means that the two series are always equal, no matter what. So, the original comparison test would always return the same answer, no matter what x was.
roam
Find the interval of convergence of the given series and its behavior at the endpoints:

$$\sum^{+\infty}_{n=1} \frac{(x+1)^n}{\sqrt{n}}$$ $$= (x+1) + \frac{(x+1)^2}{\sqrt(2)}+...$$

The attempt at a solution

Using the ratio test: $$\left|\frac{S_{n+1}}{S_{n}}\right| = \sqrt{\frac{n}{n+1}}\left|x+1\right|$$

Hence, $$lim_{n\rightarrow\infty} \left|\frac{S_{n+1}}{S_{n}}\right|= \left|x+1\right|$$

So, the interval of convergence is |x+1|<1, which implies -1<x<1 which in turn is equal to: -2<x<0

At the right hand endpoint where x = 0 we have the divergent p-series $$\sum^{+\infty}_{n=1} \frac{1}{\sqrt{n}}$$ (with p=1/2).

At the left endpoint x=-2 we get the alternating series $$\sum^{+\infty}_{n=1} \frac{(-1)^n}{\sqrt{n}}$$. The book says this series converges but I used the comparison test & found out that it's NOT!

Sn = (1)n+11/√n diverges by comparison with the p-series ∑1/√n (since p<1)

I’m really confused right now, if it is supposed to converge then why does it diverge by the comparison test? What mistakes did I make?

Thanks.

I'm confused by what you are comparing $$\frac{(-1)^n}{\sqrt{n}}$$ to?

How can you compare it to $$\frac{1}{\sqrt{n}}$$?

The first has negative and positive terms whereas the latter does not.

NoMoreExams said:
I'm confused by what you are comparing $$\frac{(-1)^n}{\sqrt{n}}$$ to?

How can you compare it to $$\frac{1}{\sqrt{n}}$$?

The first has negative and positive terms whereas the latter does not.

So, what else can I compare it to...?

Why can't we just re-write the first as $$(-1)^{n+1}\frac{1}{\sqrt{n}}$$ and then compare it to the p-series $$\sum \frac{1}{\sqrt{n}}$$?

because $$(-1)^{n+1} \frac{1}{\sqrt{n}}$$ will still have positive and negative terms right? You'd need a test that has "Absolute" in the name :) if anything.

Hmm, I'm really curious!...
For example, what test can you use for $$\frac{(-1)^n}{\sqrt{n}}$$?

Roam

roam said:
Hmm, I'm really curious!...
For example, what test can you use for $$\frac{(-1)^n}{\sqrt{n}}$$?

Roam
How about the alternating series test? You know that one, don't you?

Oh OK, here's a part of the theorem for alternating series: Let ∑(-1)n+1an be an alternating series. If the sequance $$\left\langle a_{n}\right\rangle$$ is decreasing then ∑(-1)n+1an converges to a sum A.

So, for $$\frac{(-1)^n}{\sqrt{n}}$$, we can write it as $$(-1)^n \frac{1}{\sqrt{n}}$$.

Since $$\left\langle \frac{1}{\sqrt{n}} \right\rangle$$ is a decreasing sequance (& also divergent), the series converge by virtue of the alternating series test.

Is this correct? Does it make any sense now?

Makes sense to me, as an extreme example of why your original logic doesn't work, think about what the following will produce:

$$\sum_{n=1}^{\infty} \frac{(-1)^n}{n}$$ vs. $$\sum_{n=1}^{\infty} \frac{(1)^n}{n}$$

The fact that you have alternating terms is a pretty big deal.

## What is the definition of "Interval of Convergence" for a series?

The Interval of Convergence for a series is the range of values for which the series will converge, or approach a finite sum. It is typically expressed in terms of the variable, x, and can be found by applying various convergence tests to the series.

## What is the formula for finding the Interval of Convergence for a power series?

The formula for finding the Interval of Convergence for a power series is R = 1/L, where L is the limit of the ratio of consecutive terms in the series. This formula is derived from the Ratio Test, which is commonly used to determine the convergence of power series.

## What are some common convergence tests used to find the Interval of Convergence for a series?

Some common convergence tests used to find the Interval of Convergence for a series include the Ratio Test, the Root Test, and the Comparison Test. Other tests such as the Integral Test, the Alternating Series Test, and the Direct Comparison Test may also be used depending on the type of series.

## How does the Interval of Convergence relate to the Radius of Convergence for a power series?

The Interval of Convergence and the Radius of Convergence are closely related concepts. The Radius of Convergence refers to the distance from the center of the power series to the nearest point where the series converges. The Interval of Convergence is the range of values for which the series will converge, and it can be thought of as the interval centered at the center of the power series with a length equal to the Radius of Convergence.

## What happens if the Limit of the Ratio Test is equal to zero when finding the Interval of Convergence?

If the Limit of the Ratio Test is equal to zero, it means that the series will converge for all values of x. This is known as a power series with an infinite Radius of Convergence, and the Interval of Convergence would be (-∞, ∞). In this case, the series is said to converge absolutely, meaning that the sum of the series is finite for all values of x.

• Calculus and Beyond Homework Help
Replies
2
Views
488
• Calculus and Beyond Homework Help
Replies
3
Views
364
• Calculus and Beyond Homework Help
Replies
2
Views
987
• Calculus and Beyond Homework Help
Replies
1
Views
590
• Calculus and Beyond Homework Help
Replies
17
Views
1K
• Calculus and Beyond Homework Help
Replies
6
Views
510
• Calculus and Beyond Homework Help
Replies
14
Views
1K
• Calculus and Beyond Homework Help
Replies
3
Views
708
• Calculus and Beyond Homework Help
Replies
4
Views
659
• Calculus and Beyond Homework Help
Replies
4
Views
1K