Chapter 7: Q. 2 (page 654)

A series of monomials: Use the ratio test for absolute convergence to find all values of x for which the series $\sum_{k = 1}^{\infty} \frac{x^{k}}{k!}$ converges.

Short Answer

Expert verified

The value of x is less than k

Step by step solution

Step 1. Given

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

Step 2. Ratio test

$According to the RatioTest, {\sum a_{k}}_{k = 1}^{\infty} be the series with positive terms, if L = \lim_{k \to \infty} \frac{a_{k + 1}}{a_{k}} If aL < 1 series converges . 2 . If L > 1 series diverges . 3 . If L = 1 the test is inconclusive .$

Step 3. Finding the value of x.

$Now, calculate the value of Now, calculate the value of \frac{a_{k + 1}}{a_{k}} Consider the general term a s = \frac{x^{k}}{(k)!} . Therefore, by substituting k = k + 1 a_{k + 1} = \frac{x^{(k + 1)}}{(k + 1)!} N o w, \frac{a_{k + 1}}{a_{k}} = \frac{\frac{x^{k + 1}}{(k + 1)!}}{x^{k} / k!} Using formula n! = n (n - 1)! \frac{a_{k + 1}}{a_{k}} = \frac{x}{k} T a k i n g l i m i t s \lim_{k \to \infty} \frac{a_{k + 1}}{a_{k}} = \lim_{k \to \infty} (\frac{x}{k}) x < k . for all values of x less than k, the series is convergent$

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A series of monomials: Use the ratio test for absolute convergence to find all values of x for which the series $\sum_{k = 1}^{\infty} \frac{x^{k}}{k!}$ converges.

Short Answer

Step by step solution

Step 1. Given

Step 2. Ratio test

Step 3. Finding the value of x.

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A series of monomials: Use the ratio test for absolute convergence to find all values of x for which the series∑k=1∞xkk!converges.

Short Answer

Step by step solution

Step 1. Given

Step 2. Ratio test

Step 3. Finding the value of x.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Mechanics Maths

Decision Maths

Statistics

Theoretical and Mathematical Physics

Geometry

Study anywhere. Anytime. Across all devices.

A series of monomials: Use the ratio test for absolute convergence to find all values of x for which the series $\sum_{k = 1}^{\infty} \frac{x^{k}}{k!}$ converges.