Video: Identifying the Series That Are Not Equivalent to a Given Series

Express the series βˆ‘_(𝑛 = 1)^(∞) (𝑛 + 1)/𝑛⁴ as a series that starts at 𝑛 = 3.

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Video Transcript

Which of the following series is not equivalent to the sum from 𝑛 equals one to ∞ of 𝑏 𝑛? 𝑏 one add 𝑏 two add the sum from 𝑛 equals zero to ∞ of 𝑏 𝑛 add three. 𝑏 one add the sum from 𝑛 equals two to ∞ of 𝑏 𝑛. The sum from 𝑛 equals zero to ∞ of 𝑏 𝑛 add one. 𝑏 one add 𝑏 two add the sum from 𝑛 equals one to ∞ of 𝑏 𝑛 add three. Or 𝑏 one add 𝑏 two add the sum from 𝑛 equals three to ∞ of 𝑏 𝑛.

Let’s start by having a think about what this series looks like. The first term of the series, the sum from 𝑛 equals one to ∞ of 𝑏 𝑛, will be 𝑏 one because our starting value is 𝑛 equals one. And our second term will be 𝑏 two and our third term will be 𝑏 three. And it will continue in this way. So let’s have a look at each of the five series that we’ve been given. For this first series, we have 𝑏 one add 𝑏 two. And then we’re adding the series the sum from 𝑛 equals zero to ∞ of 𝑏 𝑛 add three. So the starting value of 𝑛 is zero. So the first term of the series will be 𝑏 zero add three, which is just 𝑏 three.

The next term will be when 𝑛 equals one. So this will be 𝑏 one add three. So this will be 𝑏 four. And as this goes up to ∞, this will just continue in this way. For the second series, this starts with 𝑏 one and then we have the series the sum from 𝑛 equals two to ∞ of 𝑏 𝑛. So the first value will be when 𝑛 equals two. So this will be 𝑏 two. And then the next value will be 𝑏 three and again this goes up to ∞. So this will continue in this way.

The third series is the sum from 𝑛 equals zero to ∞ of 𝑏 𝑛 add one. So the first term will be when 𝑛 equals zero. So we’ll have 𝑏 zero add one, which is just 𝑏 one. Then when 𝑛 equals one, we’ll have 𝑏 two. And this will continue as the last value of 𝑛 is ∞. If we then have a look at the fourth series here, we have 𝑏 one add 𝑏 two. And then we have the series the sum from 𝑛 equals one to ∞ of 𝑏 𝑛 add three. So this series starts when 𝑛 equals one. And if we substitute that into the formula for the 𝑛th term, we have 𝑏 one add three, which gives us 𝑏 four. And then when 𝑛 equals two, we’ll have 𝑏 two add three, which is 𝑏 five. And the last value of 𝑛 is ∞. So this will continue in this way.

So as we’ve missed out 𝑏 three, it looks like this one is not equivalent. But let’s double check the last series to be sure. This series starts with 𝑏 one add 𝑏 two and then we have the series the sum from 𝑛 equals three to ∞ of 𝑏 𝑛. So when 𝑛 equals three, we have 𝑏 three; when 𝑛 equals four, we have 𝑏 four; and so on. So we can say that the series which is not equivalent to the sum from 𝑛 equals one to ∞ of 𝑏 𝑛 is 𝑏 one add 𝑏 two add the sum from 𝑛 equals one to ∞ of 𝑏 𝑛 add three.

So we’ve seen some infinite series which sum to ∞ and some which sum to a finite number. And in fact, there’s two really important words in mathematics which we use to describe series, convergent and divergent. A convergent series is one where the value of the series is a finite number, such as the series we saw earlier which gave us the value of one. A divergent series is one where we say the value of the series is ∞. It could be the positive or negative ∞, such as the series the sum from 𝑛 equals one to ∞ of 𝑛.

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