write a power series for f(x) = 3/(2-4x). write the series in closed form. give the interval of convergence.
3/(2-4x)
= (3/2)[1/(1-2x)]
sum n=o to oo (3/2)(2^n)(x^n)
how do i get the interval of convergence?
$\displaystyle \frac {3}{2} \sum_{n=0}^{ \infty} (2x)^n$
We need not consider the constant $\displaystyle \frac {3}{2} $
By the Root Test, we have:
$\displaystyle \lim_{n \to \infty} \left| (2x)^n \right|^{ \frac {1}{n}} = \lim_{n \to \infty} \left|2x \right|$
The series converges for $\displaystyle \left| 2x \right| = 2 \left| x \right| < 1 \Rightarrow |x| < \frac {1}{2}$
So the radius of convergence is $\displaystyle \frac {1}{2}$
Test the endpoints:
If $\displaystyle x = \frac {1}{2}$
$\displaystyle \Rightarrow \sum (2x)^n = \sum 1^n$ which diverges by the test for divergence
If $\displaystyle x = - \frac {1}{2}$
$\displaystyle \Rightarrow \sum (2x)^n = \sum (-1)^n$ which diverges by the test for divergence
so the Interval of convergence is $\displaystyle \left( - \frac {1}{2} , \frac {1}{2} \right)$
i used the root test, you always put an exponent 1/n.
you would get the same result using the ratio test, but since the exponent of 2x was just n, i decided to use the root test to get rid of the power, instead of going through the root test and having to form a fraction and all that stuff
the ratio test will give the same result, now that you know the procedure to find the IOC, try it using the ratio test
Here is a weaker version of the root canal test. The strong version is based on the notion of a "limsup" which is not convered in any Calculus course.
Given a series $\displaystyle \sum_{n=1}^{\infty}a_n$. And given the limit $\displaystyle L = \lim |a_n|^{1/n}$ (if it exists or infinite). We have the following cases:
If $\displaystyle L<1$ then the series converges absolutely.
If $\displaystyle L>1 \mbox{ or }L=+\infty$ then the series diverges.
If $\displaystyle L=1$ the test doth not work.
This is a beautiful test, which should be taken advantage of as much as possible. I like it more than the ratio test. (In fact it is stronger!)
except in advanced calculus
lol, why call it the "root canal" test then?
This is a beautiful test, which should be taken advantage of as much as possible. I like it more than the ratio test. (In fact it is stronger!)
i always have trouble applying it when factorials are involved, i think i did it once though.