I am finding the radius of convergence and interval of convergence.
the sum of n=1 to infinity of x^n/(6^n*n^4)
i used the ratio test on this and found that the limit goes to 1 and the radius of convergence is 6.
when x is -6, i know the series alternates but when i compare b of n to b of n+1 I don't see how b of n is greater than b of n+1 which is one of the stipulations for the alternating series test.
Hello, aaronb!
Interesting . . . I've seen the Ratio Test described in several ways now.
. . And a few of them never form a ratio.
Find the radius of convergence and interval of convergence:
. .
. .
Then: .
So we have: .
. . The radius of convergence is: .
At an alternating -series which converges
At a convergent -series
. . The interval of convergence is: .
I don't know what you mean by that. Do you mean that the limit of the ratios is less than 1 for |x|< 6?
Yes, that is correct.and the radius of convergence is 6.
When x= -6, this iswhen x is -6, i know the series alternates but when i compare b of n to b of n+1 I don't see how b of n is greater than b of n+1 which is one of the stipulations for the alternating series test.
It should be clear that that is larger than 6 so the numerator is larger than the denominator- the sequence is decreasing.