# How do you test for convergence for #sum(5^k+k)/(k!+3)# from k=1 to infinity?

See below.

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To test for convergence of the series (\sum_{k=1}^\infty \frac{5^k+k}{k!+3}), you can use the ratio test.

The ratio test states that if (\lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right|) is less than 1, then the series converges. If it's greater than 1, the series diverges. If it equals 1, the test is inconclusive.

For the given series:

[a_k = \frac{5^k+k}{k!+3}]

Then, applying the ratio test:

[\lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right| = \lim_{k \to \infty} \left| \frac{\frac{5^{k+1}+(k+1)}{(k+1)!+3}}{\frac{5^k+k}{k!+3}} \right|]

[= \lim_{k \to \infty} \left| \frac{5^{k+1}+(k+1)}{(k+1)!+3} \times \frac{k!+3}{5^k+k} \right|]

[= \lim_{k \to \infty} \left| \frac{5^{k+1}+(k+1)}{5^k+k} \times \frac{k!+3}{(k+1)!+3} \right|]

[= \lim_{k \to \infty} \left| \frac{5^{k+1}+(k+1)}{5^k+k} \times \frac{1}{k+1} \right|]

[= \lim_{k \to \infty} \left| \frac{5(5^k)+(k+1)}{5^k+k} \times \frac{1}{k+1} \right|]

[= \lim_{k \to \infty} \left| \frac{5+\frac{k+1}{5^k}}{1+\frac{k}{5^k}} \right|]

As (k \to \infty), (\frac{k+1}{5^k}) and (\frac{k}{5^k}) both tend to 0, thus:

[= \left| \frac{5+0}{1+0} \right| = 5]

Since the limit is greater than 1, by the ratio test, the series diverges.

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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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