How do you test the series #Sigma sqrt(n+1)sqrtn# from n is #[0,oo)# for convergence?
To test the series (\sum_{n=0}^{\infty} (\sqrt{n+1}  \sqrt{n})) for convergence, you can use the telescoping series test.
Here's how:

First, express each term as a difference of squares to simplify the series.

Observe the pattern of cancellation to determine if the series converges or diverges.
Let's proceed:

Express each term as a difference of squares: [\sqrt{n+1}  \sqrt{n} = (\sqrt{n+1}  \sqrt{n}) \times \frac{\sqrt{n+1} + \sqrt{n}}{\sqrt{n+1} + \sqrt{n}} = \frac{(\sqrt{n+1})^2  (\sqrt{n})^2}{\sqrt{n+1} + \sqrt{n}} = \frac{(n+1)  n}{\sqrt{n+1} + \sqrt{n}} = \frac{1}{\sqrt{n+1} + \sqrt{n}}]

Now, observe the pattern of cancellation: [\frac{1}{\sqrt{n+1} + \sqrt{n}} = \frac{1}{\sqrt{n+1} + \sqrt{n}} \times \frac{\sqrt{n+1}  \sqrt{n}}{\sqrt{n+1}  \sqrt{n}} = \frac{\sqrt{n+1}  \sqrt{n}}{(n+1)  n} = \sqrt{n+1}  \sqrt{n}]
Notice that each term cancels out with the next term, leaving only the first term (\sqrt{1}  \sqrt{0} = 1).
Since the series reduces to a finite value (1), the series (\sum_{n=0}^{\infty} (\sqrt{n+1}  \sqrt{n})) converges.
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The series:
is divergent.
We can see that the series is divergent by analyzing the partial sums:
so that:
We can however use a convergence test in the following way: we have that:
Using the identity:
this becomes:
Now we can use the limit comparison test with the harmonic series:
which proves that:
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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