Power Series(Radius and interval of convergence)

In summary, the conversation discusses finding the radius of convergence for a given question. The individual has already determined the radius to be 1/8 and explains that they used the ratio test to find it. They then confirm with the other participant that the series converges for all x and no need to check at infinity.
  • #1
naspek
181
0
hey there..
i've try this question(attachment) already...
i got radius of convergence, R = 1/8

at x = 1/8,
converges...

at x = -1/8
i don't how to do this one..
please help me..
 

Attachments

  • power series.bmp
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  • #2
How did you conclude that has radius of convergence 1/8?? Your bmp has an n! in the denominator. What does the ratio test tell you?
 
  • #3
Dick said:
How did you conclude that has radius of convergence 1/8?? Your bmp has an n! in the denominator. What does the ratio test tell you?

to find the radius, r..
i use an / an+1
 
  • #4
naspek said:
to find the radius, r..
i use an / an+1

Go ahead. Use it. What the limit of the ratio?
 
  • #5
then.. i got radius = 1/8
 
  • #6
naspek said:
then.. i got radius = 1/8

What's the ratio an/an+1? I don't think you are getting that limit right.
 
  • #7
i've revised back my calculation..
should i get infinity for my radius..
 
  • #8
naspek said:
i've revised back my calculation..
should i get infinity for my radius..

Yes.
 
  • #9
ok.. so.. i need to check the interval of convergence..

when x = infinity,
the series will converges..

when c = -infinity,
the series will converges too..

hence, my interval of convergence is (-infinity,infinity)

am i right?
 
  • #10
It converges for ALL x, that's what you mean, right? There's no such number as +infinity or -infinity. They are only limits you approach. You don't have to check them.
 
  • #11
alright! i understand now..
Thanks Dick =)
 

Related to Power Series(Radius and interval of convergence)

What is a power series?

A power series is a mathematical series in the form of a sum of terms where each term is a constant multiplied by a variable raised to a non-negative integer power.

What is the radius of convergence of a power series?

The radius of convergence of a power series is the distance from the center of the series to the nearest point where the series still converges. It is denoted by the letter R and can be determined using the ratio test.

How is the interval of convergence of a power series determined?

The interval of convergence of a power series is determined by finding the values of the variable for which the series converges. This can be done by using the ratio test and checking the endpoints of the interval.

Can the radius of convergence of a power series be negative?

No, the radius of convergence of a power series cannot be negative. It is always a positive value or infinity.

What is the significance of the radius and interval of convergence in a power series?

The radius and interval of convergence tell us the range of values for which the power series is valid and converges. This information is important in determining the applicability and accuracy of a power series in a given situation.

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