Answered: 2k + 1 E(-1)*+1- k! k=1

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section10.5: The Binomial Theorem

Problem 16E

See similar textbooks

Related questions

Question

Determine whether the following series converge. Justify your answers.

Expert Solution

Step 1

Given:

$\sum_{k = 1}^{\infty} {(- 1)}^{k + 1} \frac{2 k + 1}{k!}$

To Determine:

Whether the given series converges or not.

Concept:

$\begin{array}{l} (a) . Alternating Series Test: \\ If the series \sum a_{n}, where a_{n} = {(- 1)}^{n} b_{n} or a_{n} = {(- 1)}^{n + 1} b_{n}, where b_{n} \geq 0 \forall n . Then if; \\ (i) . \lim_{n \to \infty} b_{n} = 0 and \\ (i i) . \{b_{n}\} is a decreasing sequence \\ the series \sum a_{n} is convergent. \end{array}$

$\begin{array}{l} (b) . Let \{x_{n} = x (n)\} be any sequence, then x_{n} is a decreasing if \\ for n_{1} \leq n_{2}, x_{n_{1}} \geq x_{n_{2}} . \end{array}$

Step 2

Explanation:

$\begin{array}{l} \sum_{k = 1}^{\infty} {(- 1)}^{k + 1} \frac{2 k + 1}{k!} \\ \Rightarrow b_{k} = \frac{2 k + 1}{k!} \\ b_{k} \geq 0, for all values of k = 1, 2, . . . . . \\ (i) . \lim_{k \to \infty} b_{k} = \lim_{k \to \infty} \frac{2 k + 1}{k!} = 0 \\ (i i) . Now, we have to show b_{k} is a decreasing sequence \\ let k_{1} \leq k_{2} \\ \Rightarrow k_{1}! \leq k_{2}! \\ \Rightarrow \frac{1}{k_{1}!} \geq \frac{1}{k_{2}!} \\ \Rightarrow \frac{2 k_{1} + 1}{k_{1}!} \geq \frac{2 k_{2} + 1}{k_{2}!} \\ \Rightarrow b_{k_{1}} \geq b_{k_{2}} \\ \Rightarrow b_{k} is a decreasing sequence. \end{array}$

$∴ By alternating series test \sum_{k = 1}^{\infty} {(- 1)}^{k + 1} \frac{2 k + 1}{k!} converges.$

Step by step

Solved in 3 steps

Knowledge Booster

$Series$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.

Recommended textbooks for you

Algebra & Trigonometry with Analytic Geometry

Algebra

ISBN:

9781133382119

Author:

Swokowski

Publisher:

Cengage