00:01
Consider the following series, the sum of x to the power of ln of n for n ranging between 2 and infinity.
00:08
In question a, we want to determine the convergence or divergence of our series if x is equal to 1.
00:16
So when x is equal to 1, our series will be rewrite to the sum of 1 to the power of n for n ranging between 2 and infinity.
00:29
And 1 to the power of anything will always return 1.
00:33
So this simply rewrites as the sum over 1.
00:39
So essentially in this series, we are summing an infinite number of 1s and this diverges.
00:46
Next, we want to determine the convergence of our series if x is equal to 1 over e.
00:56
So if x is equal to 1 over e, we now have the sum of 1 over e to the power of ln of n for n ranging between 2 and infinity.
01:16
So let's distribute our ln of n.
01:18
We will have the sum of 1 over e to the ln of n for n ranging between 2 and infinity.
01:28
And there's a property of the exponentials that e to the ln of n is simply n.
01:37
So when x is equal to 1 over e, we are in fact summing 1 over n.
01:47
And this series is known as a harmonic series and is known to diverge.
01:56
So when x is equal to 1 over e, our series diverges.
02:04
Now we want to determine the interval of convergence of our series.
02:10
So previously when x is equal to 1 over e, we had obtained a harmonic series.
02:13
And harmonic series is very close to a p -series, but when p is equal to 1.
02:18
So let's try to rewrite our sum as a p -series...