Answered: 2* 00 k! k=1

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter1: Fundamental Concepts Of Algebra

Section1.2: Exponents And Radicals

Problem 46E

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Question

write formal test names and conditions before using the test

proper notation

proper conclusion

show algebraic simplification

Expert Solution

Given,

$\sum_{k = 1}^{\infty} \frac{2^{k}}{k!}$

Here,

$a_{k} = \frac{2^{k}}{k!}$

Let us use Ratio test to check the convergence of the given series.

Now consider,

$\begin{array}{rcl} \lim_{k \to \infty} |\frac{a_{k + 1}}{a_{k}}| & = & \lim_{k \to \infty} |\frac{\frac{2^{k + 1}}{(k + 1)!}}{\frac{2^{k}}{k!}}| \\ = & \lim_{k \to \infty} |\frac{2^{k + 1} \cdot k!}{2^{k} \cdot (k + 1)!}| \\ = & \lim_{k \to \infty} |\frac{2^{k} \cdot 2 \cdot k!}{2^{k} \cdot (k + 1) \cdot k!}| \\ = & \lim_{k \to \infty} |\frac{2}{k + 1}| \\ = & \frac{2}{\infty} \\ = & 0 < 1 \end{array}$

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