How do you test the series #sum_(n=0)^(oo) n/((n+1)(n+2))# for convergence?
To test the series ( \sum_{n=0}^{\infty} \frac{n}{(n+1)(n+2)} ) for convergence, we can use the ratio test.

Calculate ( \lim_{n \to \infty} \frac{a_{n+1}}{a_n} ), where ( a_n = \frac{n}{(n+1)(n+2)} ).

If the limit is less than 1, the series converges absolutely. If it's greater than 1 or the limit doesn't exist, the series diverges. If the limit equals 1, the ratio test is inconclusive.
Let's apply the ratio test:
[ \lim_{n \to \infty} \frac{\frac{n+1}{((n+1)+1)((n+1)+2)}}{\frac{n}{(n+1)(n+2)}} ]
[ = \lim_{n \to \infty} \frac{n+1}{(n+2)(n+3)} \times \frac{(n+1)(n+2)}{n} ]
[ = \lim_{n \to \infty} \frac{n+1}{n} \times \frac{n+1}{n+3} ]
[ = \lim_{n \to \infty} \frac{(n+1)^2}{n(n+3)} ]
[ = \lim_{n \to \infty} \frac{n^2 + 2n + 1}{n^2 + 3n} ]
[ = 1 ]
Since the limit equals 1, the ratio test is inconclusive. Therefore, we need to use another test, such as the divergence test or comparison test, to determine the convergence or divergence of the series.
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The series diverge
Perform the limit comparison test
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The series:
is divergent.
The series has only positive terms, so we can use the limit comparison test to compare it with the harmonic series:
As the limit is finite and positive the two series have the same character, and we know the harmonic series to be divergent, thus also the series:
is divergent.
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We can use the integral test to find it diverges.
Using the integral test, we find:
So:
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See below.
Now considering
know as divergent.
Resuming
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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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