00:01
And so in this question, they say, i want to find the interval of convergence of the power series, and we're going to check for convergence at the end points of our interval.
00:09
Here i have the sum from n equals 1 to infinity of 4 to the n power times the quantity of x minus 1 to the n power over n.
00:18
And so if i want to find an interval of convergence, what do i do? i use the ratio test.
00:25
I know my series converges if the limit has an approached infinity, of the absolute value of a sub n plus 1 over a sub n is less than 1.
00:39
Now in this case, my a sub n plus 1 is 4 to the n plus 1 times x minus 1 to the n plus power over n plus 1.
00:52
That's a subn plus 1.
00:54
Now i divide by a sub n.
00:57
I am dividing by n over 4 to the n power times x minus 1.
01:05
1 to the n power, and i need this to be less than 1.
01:13
Let's see, i have a limit as an approach to infinity of the absolute value of light.
01:19
Now, 4 to the n plus 1 over 4 to the n, that simplifies as 4 up top.
01:25
X minus 1 to the n plus 1 over x minus 1 to the end.
01:30
That simplifies as x minus 1 up top.
01:34
I have an n up top and an n plus 1 in the bottom, and i need all of this to be less than 1.
01:43
Now, this is a limit as n approaches infinity, which means i can think of that four times the quantity of x minus 1 as just a constant tier.
01:56
And so, therefore, since this is a limit as n approaches infinity, i can pull this through my limit symbol.
02:04
The absolute value of four times the quantity of x minus one times the limit as n approaches infinity of the absolute value of n over n plus one is less than one now what's this limit this limit is one okay so what do we have the absolute value of four times the quantity of x minus one is less than what does this mean if i break this out of absolute value bars it means that four times the quantity of x minus 1 lives between negative 1 and 1.
02:48
Divide by 4, i'm getting negative 1 1 4th is less than x minus 1 is less than 1 4th.
02:59
Add 1 to all three portions of this inequality.
03:03
3 4ths is less than x, which is less than 5 quarters.
03:08
And now i've got to check my end points...