# How do you use the ratio test to test the convergence of the series #∑k/(3+k^2) # from k=1 to infinity?

Let:

and evaluate the ratio:

We have that:

so the ration test is in effect inconclusive to determine whether the series:

is convergent.

However if we consider the harmonic series:

which is divergent and we apply the limit comparison test, we can see that:

so, as the limit is finite, the two series have the same character and we can conclude that;

is divergent.

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To use the ratio test to test the convergence of the series (\sum \frac{k}{3+k^2}) from (k=1) to infinity, we compute the limit of the ratio of consecutive terms:

[ L = \lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right| = \lim_{k \to \infty} \left| \frac{\frac{k+1}{3+(k+1)^2}}{\frac{k}{3+k^2}} \right| ]

Simplify the expression:

[ L = \lim_{k \to \infty} \left| \frac{k+1}{3+(k+1)^2} \cdot \frac{3+k^2}{k} \right| ]

[ L = \lim_{k \to \infty} \left| \frac{(k+1)(3+k^2)}{(3+(k+1)^2)k} \right| ]

[ L = \lim_{k \to \infty} \left| \frac{3k+k^3+3+k^2}{3k+k^2+2k+1} \right| ]

[ L = \lim_{k \to \infty} \left| \frac{k^3+ k^2 + 3k + 3}{k^2+ 2k + 3} \right| ]

[ L = \lim_{k \to \infty} \left| \frac{k^3/k^2+ k^2/k^2 + 3k/k^2 + 3/k^2}{k^2/k^2+ 2k/k^2 + 3/k^2} \right| ]

[ L = \lim_{k \to \infty} \left| \frac{1 + 1/k + 3/k^2 + 3/k^3}{1 + 2/k + 3/k^2} \right| ]

[ L = 1 ]

Since (L = 1), the ratio test is inconclusive. We cannot determine the convergence or divergence of the series using this test alone. Additional tests may be needed.

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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.

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