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

# Question Video: Identifying the Series That Are Not Equivalent to a Given Series Mathematics • Higher Education

Express the series β_(π = 1)^(β) (π + 1)/πβ΄ as a series that starts at π = 3.

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### Video Transcript

Which of the following series is not equivalent to the sum from π equals one to β of π π? π one add π two add the sum from π equals zero to β of π π add three. π one add the sum from π equals two to β of π π. The sum from π equals zero to β of π π add one. π one add π two add the sum from π equals one to β of π π add three. Or π one add π two add the sum from π equals three to β of π π.

Letβs start by having a think about what this series looks like. The first term of the series, the sum from π equals one to β of π π, will be π one because our starting value is π equals one. And our second term will be π two and our third term will be π three. And it will continue in this way. So letβs have a look at each of the five series that weβve been given. For this first series, we have π one add π two. And then weβre adding the series the sum from π equals zero to β of π π add three. So the starting value of π is zero. So the first term of the series will be π zero add three, which is just π three.

The next term will be when π equals one. So this will be π one add three. So this will be π four. And as this goes up to β, this will just continue in this way. For the second series, this starts with π one and then we have the series the sum from π equals two to β of π π. So the first value will be when π equals two. So this will be π two. And then the next value will be π three and again this goes up to β. So this will continue in this way.

The third series is the sum from π equals zero to β of π π add one. So the first term will be when π equals zero. So weβll have π zero add one, which is just π one. Then when π equals one, weβll have π two. And this will continue as the last value of π is β. If we then have a look at the fourth series here, we have π one add π two. And then we have the series the sum from π equals one to β of π π add three. So this series starts when π equals one. And if we substitute that into the formula for the πth term, we have π one add three, which gives us π four. And then when π equals two, weβll have π two add three, which is π five. And the last value of π is β. So this will continue in this way.

So as weβve missed out π three, it looks like this one is not equivalent. But letβs double check the last series to be sure. This series starts with π one add π two and then we have the series the sum from π equals three to β of π π. So when π equals three, we have π three; when π equals four, we have π four; and so on. So we can say that the series which is not equivalent to the sum from π equals one to β of π π is π one add π two add the sum from π equals one to β of π π add three.

So weβve seen some infinite series which sum to β and some which sum to a finite number. And in fact, thereβ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 β. It could be the positive or negative β, such as the series the sum from π equals one to β of π.

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