Power Series(Radius and interval of convergence)

In summary, the power series \sum\limits_{k=0}^\inftyk!(x-3)^k has a radius of convergence of 0 and an interval of convergence of {3}.
  • #1
naspek
181
0
hey there.. i hope u guys can help me..
the question is...
Determine the interval and radius of convergence of the power series below..
[tex]\sum\limits_{k=0}^\infty[/tex]k!(x-3)^k

i've already find the radius, r = 0
then..
c - r < x < c + r
3 - 0 < x < 3 + 0
hence, x = 3

when x = 3..
[tex]\sum\limits_{k=0}^\infty[/tex]k!(3-3)^k = 0

so.. the interval of convergence is 0
am i do it right?
 
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  • #2
naspek said:
hey there.. i hope u guys can help me..
the question is...
Determine the interval and radius of convergence of the power series below..
[tex]\sum\limits_{k=0}^\infty[/tex]k!(x-3)^k

i've already find the radius, r = 0
then..
c - r < x < c + r
3 - 0 < x < 3 + 0
hence, x = 3

when x = 3..
[tex]\sum\limits_{k=0}^\infty[/tex]k!(3-3)^k = 0

so.. the interval of convergence is 0
am i do it right?
No, the interval of convergence is an interval- a set of points, not a number. Since the radius of convergence is 0, the "interval" of convergence is the single point {3}
 
  • #3
Got it! Thanks! =)
 

What is a power series?

A power series is a type of infinite series that represents a mathematical function as a sum of terms that are multiples of powers of a variable. It is written in the form of Σ anxn, where a is a constant coefficient and x is the variable.

What is the radius of convergence of a power series?

The radius of convergence of a power series is a non-negative number that represents the distance from the center point of the series (where x = 0) to the point where the series converges. This means that the series will only converge within this radius and will diverge outside of it.

How do you find the radius of convergence of a power series?

To find the radius of convergence, you can use the ratio test or the root test. These tests involve taking the limit of the ratio or root of the absolute value of the terms in the series. If the limit is less than 1, then the series will converge. The radius of convergence can then be found by taking the reciprocal of this limit.

What is the interval of convergence of a power series?

The interval of convergence of a power series is the range of values for the variable x for which the series converges. This interval may be a single point, a finite interval, or an infinite interval.

How do you find the interval of convergence of a power series?

To find the interval of convergence, you can use the ratio or root test as mentioned before. Once you have found the radius of convergence, you can use this information to determine the interval of convergence. For example, if the radius of convergence is 2, then the interval of convergence will be from -2 to 2, including the endpoints.

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