Chapter 7: Q. 18 (page 639)

Explain why the series $\sum_{k = 0}^{\infty} \frac{1}{k!}$ converges. Which convergence tests could be used to prove this?

Short Answer

Expert verified

Hence proved.

Step by step solution

Step 1. Given Information.

The given series is $\sum_{k = 0}^{\infty} \frac{1}{k!}$ .

Step 2. Ratio Test.

Using the ratio test,

$\frac{a_{k + 1}}{a_{k}} = \frac{\frac{1}{(k + 1)!}}{\frac{1}{k!}} \frac{a_{k + 1}}{a_{k}} = \frac{k!}{(k + 1) k!} = \frac{1}{k + 1} \lim_{k \to \infty} \frac{a_{k + 1}}{a_{k}} = \lim_{k \to \infty} \frac{1}{(k + 1)} = 1 (\lim_{k \to \infty} \frac{1}{(k + 1)}) = 0 S i n c e, L < 1,$

Therefore, the series converges.

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Explain why the series $\sum_{k = 0}^{\infty} \frac{1}{k!}$ converges. Which convergence tests could be used to prove this?

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Ratio Test.

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Explain why the series ∑k=0∞1k!converges. Which convergence tests could be used to prove this?

Short Answer

Step by step solution

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Step 2. Ratio Test.

One App. One Place for Learning.

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Explain why the series $\sum_{k = 0}^{\infty} \frac{1}{k!}$ converges. Which convergence tests could be used to prove this?