View Full Version : Power Series - Can someone please help me understand them?
NINHARDCOREFAN
Dec14-04, 05:57 PM
I don't understand how to get the inteveral of convergence from a problem:
for example:
infinity
_
\
/ -(1)^n(x-2)^n/(n*4^n)
_
n=1
I know you have to use ratio test and it comes out to:
|x-2|/4
The textbook says the series converges when |x-2|<4. How did they come to this conclusion? How do you find c? the answer was c=2. The series converges when -2<x<6 and diverges if either x<-2 or x>6. Can someone explain this also? Thanks
fourier jr
Dec14-04, 06:27 PM
\sum_{n=1}^\infty \frac{((-1)^n)(x-2)^n}{n4^n} converges when a_n < 1 , so |x-2|/4 < 1 => |x-2|< 4.
|x-2| < 4 means that for -2 < x < 6, the series converges. plug x=-2 into the series & see that the series converges @ x=-2. plug x=6 into the series & it converges there also. so the interval of convergence is [-2, 6]
NINHARDCOREFAN
Dec14-04, 08:22 PM
You just listed the facts and didn't explain them explicitly... anyway I understood most of the stuff through doing the problems. But I still don't understand one thing, when you plug in the x how do you find the limit when n is at infinity, I'm getting confused because of n being the power...
When you plugged in x=2 how do you solve for it?
Tom Mattson
Dec14-04, 08:42 PM
I know you have to use ratio test and it comes out to:
|x-2|/4
Right.
The textbook says the series converges when |x-2|<4. How did they come to this conclusion?
They came to it by recognizing that under the ratio test a series converges absolutely if the limit as n goes to infinity of |an+1/an| is less than one. So you get:
|x-2|/4<1
or...
|x-2|<4
How do you find c? the answer was c=2.
When you work the inequality above into the form:
|x-c|<R,
then c is the center of the series and R is the radius of convergence.
The series converges when -2<x<6 and diverges if either x<-2 or x>6. Can someone explain this also? Thanks
Use the definition of the absolute value.
|x-2|<4
-4<(x-2)<4
-2<x<6
Now that doesnt' tell you the endpoint convergence. You still have to test for that. So you plug x=-2 and x=6 into your power series (one at a time, of course) and you see if the resultant infinite series converge or diverge. That will tell you which inequality to put an "equal" sign under, if any.
NINHARDCOREFAN
Dec14-04, 09:21 PM
Thanks. But you didn't answer my main question, how do I solve the series after plugging in the x value (into the series)?
Tom Mattson
Dec14-04, 09:26 PM
Thanks. But you didn't answer my main question, how do I solve the series after plugging in the x value (into the series)?
Have you not studied infinite series??
If you plug in x=-2, you get:
\sum_{n=1}^\infty \frac{((-1)^n)(-2-2)^n}{n4^n}
\sum_{n=1}^\infty \frac{((-1)^n)(-4)^n}{n4^n}
\sum_{n=1}^\infty \frac{((-1)^n)(-1)^n4^n}{n4^n}
\sum_{n=1}^\infty \frac{1}{n}
...which diverges by the integral test.
I'll leave it to you to plug in and test the series at x=6.
fourier jr
Dec14-04, 10:37 PM
doh! :frown: it's (x-2)^n not (x-2). just ignore my stuff...
You just listed the facts and didn't explain them explicitly... anyway I understood most of the stuff through doing the problems. But I still don't understand one thing, when you plug in the x how do you find the limit when n is at infinity, I'm getting confused because of n being the power...
When you plugged in x=2 how do you solve for it?
There isn't necessarily an expression for the result of a sum. People are probably working on a nice expression for
\sum_{i=1}^{\infty} \frac{1}{n^3}
right now.
All this problem is asking for is convergence, so:
The ratio between the n^{th} and (n+1)^{th} terms is
\frac{n+1}{n} \frac{(-1)(x-2)}{4}
in the limit that turns into
\frac{(-1)(x-2)}{4}
and the ratio must be less than one for the series to converge, so
\left|\frac{(-1)(x-2)}{4}\right| < 1
\left|(x-2)\right| < 4
-2 < x < 6
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