How do you test the series #Sigma (n+3)/(n(n+1)(n-2))# from n is #[3,oo)# for convergence?
To test the convergence of the series Sigma (n+3)/(n(n+1)(n-2)) from n is [3,oo), you can use the Ratio Test. Let's denote the general term of the series as a_n = (n+3)/(n(n+1)(n-2)).
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Apply the Ratio Test:
- Compute the limit as n approaches infinity of the absolute value of the ratio of consecutive terms: lim (n→∞) |a_(n+1)/a_n|.
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Simplify the expression:
- Substitute (n+1) for n in the general term to get a_(n+1), then divide a_(n+1) by a_n.
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Take the limit:
- Evaluate the limit of the ratio as n approaches infinity.
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Analyze the result:
- If the limit is less than 1, the series converges absolutely.
- If the limit is greater than 1 or undefined, the series diverges.
- If the limit equals 1, the test is inconclusive, and another test may be needed.
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If the limit is less than 1, conclude that the series converges absolutely. If the limit is greater than 1 or undefined, conclude that the series diverges.
Apply these steps to determine the convergence of the given series.
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is convergent.
Given:
We can start from:
if we increase the numerator and decrease the denominator, the resulting quotient will be bigger:
is convergent.
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When evaluating a one-sided 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 one-sided 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 one-sided 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 one-sided 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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- How do you apply the ratio test to determine if #sum_(n=2)^oo 10^n/(lnn)^n# is convergent to divergent?
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