Question Video: Determining Whether a Given Series Is Convergent or Divergent

Video Transcript

Determine whether the series the sum from 𝑛 equals one to ∞ of negative one to the 𝑛th power over 𝑛 factorial is convergent or divergent.

If we expand this sum, remembering that raising negative one to an odd power gives us negative one but raising negative one to an even power gives us one, we can then see that the terms in this series alternate between negative and positive. So, we call this an alternating series. And to determine whether an alternating series is convergent or divergent, we have a special test called the alternating series test.

The alternating series test says that an alternating series the sum from 𝑛 equals one to ∞ of negative one to the 𝑛th power multiplied by 𝑎 𝑛 converges if the limit as 𝑛 approaches ∞ of 𝑎 𝑛 equals zero and the sequence 𝑎 𝑛 is decreasing. Also, note that to use the alternating series test, we must have that 𝑎 𝑛 is nonnegative for all 𝑛.

So, if we start by recognizing that we can write our series as the sum from 𝑛 equals one to ∞ of negative one to the 𝑛th power multiplied by one over 𝑛 factorial. We can see that, for this question, 𝑎 𝑛 equals one over 𝑛 factorial. The first condition that we need to check is whether the limit as 𝑛 approaches ∞ of 𝑎 𝑛 equals zero. In other words, we want to find the limit as 𝑛 approaches ∞ of one over 𝑛 factorial.

Well, as 𝑛 increases, 𝑛 factorial will approach ∞. So, the limit of one over 𝑛 factorial will be zero. So, we’ve satisfied this first condition. So, now, we need to check whether the sequence 𝑎 𝑛 is decreasing. In other words, we must have that 𝑎 𝑛 is greater than 𝑎 𝑛 plus one.

So, is one over 𝑛 factorial greater than one over 𝑛 plus one factorial? Well, as 𝑛 factorial will always be less than 𝑛 plus one factorial for our values of 𝑛, then one over 𝑛 factorial will always be greater than one over 𝑛 plus one factorial. So, the sequence 𝑎 𝑛 must be decreasing. So, as we met both of the conditions, we can conclude that the series is convergent by the alternating series test.