How do you use the ratio test to test the convergence of the series #∑(5^k+k)/(k!+3)# from n=1 to infinity?
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To use the ratio test to test the convergence of the series ( \sum_{n=1}^{\infty} \frac{5^k + k}{k! + 3} ):

Compute the limit of the absolute value of the ratio of consecutive terms: [ \lim_{k \to \infty} \left \frac{a_{k+1}}{a_k} \right ]

If the limit is less than 1, the series converges. If it's greater than 1 or the limit does not exist, the series diverges. If the limit equals 1, the test is inconclusive.
Let's apply this to the given series:
[ a_k = \frac{5^k + k}{k! + 3} ]
[ \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! + 3)}{(5^k + k)((k+1)! + 3)} \right ]
[ = \lim_{k \to \infty} \left \frac{5^{k+1}k! + 5^{k+1} + k(k! + 3) + (k+1)(k! + 3)}{5^kk! + 5k! + k(k+1)! + 3(k+1)! + 3} \right ]
[ = \lim_{k \to \infty} \left \frac{5^{k+1}k! + 5^{k+1} + k! + 3k + k + 1}{5^kk! + 5k! + k! + 3k! + 3} \right ]
[ = \lim_{k \to \infty} \left \frac{(5^{k+1} + 1)k! + (k + 3)}{(5^k + 1)k! + (3k! + 3)} \right ]
[ = \lim_{k \to \infty} \left \frac{5^{k+1} + 1}{5^k + 1} \right = 5 ]
Since the limit is greater than 1, the series diverges by the Ratio Test.
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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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