analysis001 said:Homework Statement
Prove that the series diverges: [itex]\sum_{i=1}^{\infty}\sqrt{n+1}-\sqrt{n}[/itex]
The Attempt at a Solution
I'm trying to use the comparison test, but I don't know what to compare it to. All I have done so far is change the terms to be in fraction form:
[itex]\sqrt{n+1}[/itex]-[itex]\sqrt{n}[/itex]=[itex]\frac{1}{\sqrt{n+1}+\sqrt{n}}[/itex]
Any clues on what to do next would be great. Thanks!
analysis001 said:Homework Statement
Prove that the series diverges: [itex]\sum_{i=1}^{\infty}\sqrt{n+1}-\sqrt{n}[/itex]
The Attempt at a Solution
I'm trying to use the comparison test, but I don't know what to compare it to. All I have done so far is change the terms to be in fraction form:
[itex]\sqrt{n+1}[/itex]-[itex]\sqrt{n}[/itex]=[itex]\frac{1}{\sqrt{n+1}+\sqrt{n}}[/itex]
Any clues on what to do next would be great. Thanks!
A "Divergent Series Proof" is a mathematical proof that shows a series (a sequence of numbers) does not have a finite sum. This means that as more terms are added to the series, the sum of the terms will continue to increase without ever reaching a specific value.
In a regular proof, the goal is to show that a statement or equation is true. In a "Divergent Series Proof", the goal is to show that a series does not have a finite sum, which is considered a counterexample to the statement being true.
An example of a "Divergent Series Proof" is the proof that the harmonic series, 1 + 1/2 + 1/3 + 1/4 + ..., does not have a finite sum. This was first shown by the mathematician Nicole Oresme in the 14th century.
"Divergent Series Proofs" are important in mathematics because they demonstrate the limitations of certain mathematical concepts. They also open up new areas of study and research, as mathematicians seek to better understand these infinite series.
No, a "Divergent Series Proof" can only show that a statement is false. If a series is proven to be divergent, it does not necessarily mean that the statement is true. It only means that the statement is not always true, as there may be exceptions where the series does have a finite sum.