How do we determine the convergence at the endpoints for the series (x-1)^n/n^3?

In summary, the conversation is discussing the use of the root test to find the interval of convergence for a series. The speaker is confused about why the endpoints need to be tested, as the root test gives a limit of 1 for both values of x. The other person explains that a limit of 1 does not guarantee convergence, and gives examples to illustrate this point. They also clarify that the root test only guarantees convergence for |r| < 1, not |r| <= 1.
  • #1
GreenPrint
1,196
0
Hi,

I don't really need help with a problem, just having some troubles understanding something.

Find the interval of convergence of the series

sigma[n=0,inf] (x-1)^n/n^3
by the root test I got that |x-1|<1 and that 0<x<2
I than have to plug in these values (0 and 2) to see if the series converges or diverges at these endpoints... I however am confused by this. If we set x = 0 or x = 2 wouldn't the root test give us 1 and the root test states that if the limit equals one than the series converges. So what's the need of testing the end points? Like I know we have to just am not really sure why because than the root test would give us 1... I never understood why we test the end points and just sort of did so just because I was told we had to and was wondering if someone could explain this to me. I hope someone can clear up this confusion for me. Thanks for any help!
 
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  • #2
GreenPrint said:
Hi,

I don't really need help with a problem, just having some troubles understanding something.

Find the interval of convergence of the series

sigma[n=0,inf] (x-1)^n/n^3
by the root test I got that |x-1|<1 and that 0<x<2
I than have to plug in these values (0 and 2) to see if the series converges or diverges at these endpoints... I however am confused by this. If we set x = 0 or x = 2 wouldn't the root test give us 1 and the root test states that if the limit equals one than the series converges. So what's the need of testing the end points? Like I know we have to just am not really sure why because than the root test would give us 1... I never understood why we test the end points and just sort of did so just because I was told we had to and was wondering if someone could explain this to me. I hope someone can clear up this confusion for me. Thanks for any help!

Root test = 1 does NOT guarantee convergence. Look at sum_{n=1}^infinity 1. Does it converge? Does it satisfy root test = 1? Look at sum r^n /n. Root test implies convergence for |r| <= 1. Does the series converge at r = 1? At r = -1?

RGV
 
  • #3
Well like I forgot myself and had to look up on wikipedia http://en.wikipedia.org/wiki/Root_test and it says that if r=1 than the series diverges so I thought it was always |r| < 1 not |r| <= 1?
 
  • #4
If n=1 when using the root test, it means your answer is inconclusive AKA use another test (that's what RGV is hinting at).
 
  • #5
hm interesting thanks
 
  • #6
Sorry that I didn't answer your second question, if [tex]{n<1}[/tex] it will ALWAYS converge, if [tex]{n>1}[/tex] it will ALWAYS diverge! If [tex]{n<=1}[/tex] then I'm going to guess that it'll converge, but I've never seen a series that has n<=1!

This is for ratio/root test.
 

1. What are infinite series in Calculus II?

Infinite series are mathematical expressions that consist of an infinite number of terms. Each term in the series is added together to obtain a sum, which may or may not converge to a finite value.

2. What is the difference between a convergent and a divergent series?

A convergent series is one in which the sum of the terms approaches a finite value as the number of terms increases towards infinity. In contrast, a divergent series is one in which the sum of the terms either approaches infinity or does not approach any finite value.

3. How can I determine if an infinite series is convergent or divergent?

There are several tests that can be used to determine the convergence or divergence of a series, such as the comparison test, ratio test, and integral test. These tests involve comparing the given series to a known series with known convergence behavior.

4. What is the purpose of studying infinite series in Calculus II?

Infinite series are essential in calculus as they provide a way to approximate functions and solve problems that cannot be solved using basic algebra or calculus. They also have real-world applications in fields such as physics, engineering, and economics.

5. Are there any real-life examples of infinite series?

Yes, there are many real-life examples of infinite series. For instance, the value of pi can be approximated using an infinite series. In finance, compound interest can be represented as an infinite series. In physics, the motion of a pendulum can be described using an infinite series. These are just a few examples of the many real-life applications of infinite series.

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