Video Transcript
Is the series the sum from 𝑛
equals one to ∞ of negative one to the power of 𝑛 add one over 𝑛 squared
absolutely convergent?
Remember to test for absolute
convergence, we need to check whether the series of absolute values is convergent,
in other words, is the sum from 𝑛 equals one to ∞ of the absolute value of negative
one to the power of 𝑛 add one over 𝑛 squared convergent. First of all, notice that negative
one to the power of 𝑛 add one is always going to be either one or negative one,
depending on whether the power is even or odd. So if we take the absolute value of
negative one raised to the power of 𝑛 add one, this is always going to be one. We know the 𝑛 runs from one to
∞. So 𝑛 squared is always going to be
positive. So we can actually rewrite this as
one over 𝑛 squared.
Remember that we’re trying to
determine whether this converges or diverges. But we actually recognize the sum
from 𝑛 equals one to ∞ of one over 𝑛 squared to be a series that we know. It’s a 𝑃-series. So we use the fact that a 𝑃-series
converges if 𝑃 is greater than one and diverges if 𝑃 is less than or equal to
one. So for our question, we see that
this is a 𝑃-series with 𝑃 equal to two. Because this is greater than one,
we can say that the sum from 𝑛 equals one to ∞ of one over 𝑛 squared is
convergent. So because we found the series of
absolute values to be convergent, then the series is absolutely convergent.
Interestingly, if we find a series,
which is not absolutely convergent, it may still be convergent. We call this conditional
convergence. A series is conditionally
convergent if the series is convergent but not absolutely convergent. In other words, the sum from 𝑛
equals one to ∞ of the absolute value of 𝑎 𝑛 diverges. But the sum from 𝑛 equals one to ∞
of 𝑎 𝑛 converges. And if a series is not absolutely
convergent and it’s not conditionally convergent, then it’s divergent.
