### Problem Description**Sequence Definitions:**Let $\{a_k\}$ be a sequence of real numbers and sequences $\{b_k\}$ and $\{c_k\}$ be defined by the following rules:\[b_k = \begin{cases} a_k & \text{if } a_k > 0 \\0 & \text{if } a_k \leq 0 \end{cases}\quad \text{and} \quadc_k = \begin{cases} -a_k & \text{if } a_k < 0 \\0 & \text{if } a_k \geq 0 \end{cases}\]**Statements to Prove:**(a) Prove that, for each $ k \in \mathbb{N} $, $b_k - c_k = a_k$ and $b_k + c_k = |a_k|$.(b) Prove that, if $\sum a_k$ converges conditionally, then $\sum b_k$ and $\sum c_k$ both diverge.### Explanation of Problem Components**Sequences $\{a_k\}$, $\{b_k\}$, $\{c_k\}$:**- The sequence $\{a_k\}$ is any sequence of real numbers.- The sequence $\{b_k\}$ includes only the positive values from $\{a_k\}$ and sets everything else to 0.- The sequence $\{c_k\}$ includes the negative values from $\{a_k\}$ as positive (since $c_k = -a_k$) and sets everything else to 0.### Detailed Steps**Part (a):**1. **To show $b_k - c_k = a_k$:** - If $a_k > 0$, then - $b_k = a_k$, - $c_k = 0$, - Thus, $b_k - c_k = a_k - 0 = a_k$. - If $a_k \leq 0$, then - $b_k = 0$, - $c_k = -a_k$ (note: $-a_k$ is

Name: ### Problem Description**Sequence Definitions:**Let $\{a_k\}$ be a
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Description: ### Problem Description**Sequence Definitions:**Let $\{a_k\}$ be a sequence of real numbers and sequences $\{b_k\}$ and $\{c_k\}$ be defined by the following rules:\[b_k = \begin{cases} a_k & \text{if } a_k > 0 \\0 & \text{if } a_k \leq 0 \end

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Answered: (a) Prove that, for each k E N, b - Ck…

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(a) Prove that, for each k E N, b - Ck = ak and b+ Ck = |ak]. (b) Prove that, if ak converges conditionally, then b and C both diverge.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Concept explainers

Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence. It was invented by German mathematician Carl Friedrich Gauss.

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Question

### Problem Description

**Sequence Definitions:**

Let $\{a_k\}$ be a sequence of real numbers and sequences $\{b_k\}$ and $\{c_k\}$ be defined by the following rules:

\[
b_k =
\begin{cases}
a_k & \text{if } a_k > 0 \\
0 & \text{if } a_k \leq 0
\end{cases}
\quad \text{and} \quad
c_k =
\begin{cases}
-a_k & \text{if } a_k < 0 \\
0 & \text{if } a_k \geq 0
\end{cases}
\]

**Statements to Prove:**

(a) Prove that, for each $ k \in \mathbb{N} $, $b_k - c_k = a_k$ and $b_k + c_k = |a_k|$.

(b) Prove that, if $\sum a_k$ converges conditionally, then $\sum b_k$ and $\sum c_k$ both diverge.

### Explanation of Problem Components

**Sequences $\{a_k\}$, $\{b_k\}$, $\{c_k\}$:**
- The sequence $\{a_k\}$ is any sequence of real numbers.
- The sequence $\{b_k\}$ includes only the positive values from $\{a_k\}$ and sets everything else to 0.
- The sequence $\{c_k\}$ includes the negative values from $\{a_k\}$ as positive (since $c_k = -a_k$) and sets everything else to 0.

### Detailed Steps

**Part (a):**

1. **To show $b_k - c_k = a_k$:**
- If $a_k > 0$, then
- $b_k = a_k$,
- $c_k = 0$,
- Thus, $b_k - c_k = a_k - 0 = a_k$.
- If $a_k \leq 0$, then
- $b_k = 0$,
- $c_k = -a_k$ (note: $-a_k$ is

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