How do you test the series #Sigma n/(n+4)# from n is #[0,oo)# for convergence?
To test the series ( \sum \frac{n}{n + 4} ) for convergence, you can use the Ratio Test:

Compute the limit: [ \lim_{{n \to \infty}} \left \frac{a_{n+1}}{a_n} \right ] where ( a_n = \frac{n}{n + 4} ).

Calculate ( a_{n+1} ) by replacing ( n ) with ( n + 1 ) in the formula for ( a_n ).

Simplify the ratio ( \frac{a_{n+1}}{a_n} ).

Take the limit as ( n ) approaches infinity.

If the limit is less than 1, the series ( \sum \frac{n}{n + 4} ) converges. If it's greater than 1 or the limit doesn't exist, the series diverges. If the limit equals 1, the test is inconclusive, and you may need to use another test.
Applying these steps: [ a_n = \frac{n}{n + 4} ] [ a_{n+1} = \frac{n+1}{n+1+4} = \frac{n+1}{n+5} ] [ \lim_{{n \to \infty}} \left \frac{a_{n+1}}{a_n} \right = \lim_{{n \to \infty}} \left \frac{\frac{n+1}{n+5}}{\frac{n}{n+4}} \right = \lim_{{n \to \infty}} \left \frac{n(n+4)}{(n+1)(n+5)} \right = 1 ]
Since the limit equals 1, the Ratio Test is inconclusive for this series. You may need to use another test, such as the Comparison Test or the Limit Comparison Test, to determine convergence or divergence.
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In our case:
so the series is not convergent.
As:
then the series is divergent.
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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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