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Chapter 9: Problem 41

Prove: If \(\sum_{k=1}^{\infty} a_{k}\) diverges, so does \(\sum_{k=1}^{\infty} c a_{k}\) for \(c \neq 0\).

Short Answer

Expert verified
The series \( \sum_{k=1}^{\infty} c a_{k} \) diverges if \( \sum_{k=1}^{\infty} a_{k} \) diverges and \( c \neq 0 \).

Step by step solution

01

Understand the Problem Statement

We need to prove that if the series \( \sum_{k=1}^{\infty} a_{k} \) diverges, then the series \( \sum_{k=1}^{\infty} c a_{k} \) also diverges, where \( c \) is a non-zero constant. In simpler terms, multiplying a divergent series by a non-zero constant \( c \) should not make it converge.
02

Consider the Definition of Divergence

A series \( \sum_{k=1}^{\infty} a_{k} \) diverges if the sequence of partial sums \( S_n = a_1 + a_2 + \ldots + a_n \) does not converge to any finite limit. This means that either \( S_n \to \infty \), \( S_n \to -\infty \), or \( S_n \) oscillates without settling down to a particular value.
03

Multiplication by a Constant

If we consider the series \( \sum_{k=1}^{\infty} c a_{k} \), the corresponding sequence of partial sums will be \( T_n = c(a_1 + a_2 + \ldots + a_n) = c S_n \). Here we multiply each term of the original sequence of partial sums by \( c \).
04

Analyze the Behavior of \( T_n \)

Since \( c eq 0 \), the behavior of \( T_n \) will be directly influenced by \( c \). If \( S_n \to \infty \), then \( T_n = c S_n \to \infty \) or \( T_n \to -\infty \) if \( c < 0 \). Similarly, if \( S_n \to -\infty \), then \( T_n \to \infty \) if \( c < 0 \) or \( T_n \to -\infty \) if \( c > 0 \). If \( S_n \) oscillates, then \( T_n \) will also oscillate and not converge.
05

Conclusion of Proof

Hence, in any situation where \( S_n \) is divergent, the sequence \( T_n \) will also be divergent, confirming that the series \( \sum_{k=1}^{\infty} c a_{k} \) must diverge as well, provided \( c eq 0 \). This concludes our proof.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sequence of Partial Sums
When we talk about a series diverging, one key tool we use to understand this behavior is the sequence of partial sums. For a series \( \sum_{k=1}^{\infty} a_{k} \), the partial sums are like running totals: \( S_n = a_1 + a_2 + \ldots + a_n \). Each partial sum \( S_n \) explores how the sum grows as we include more terms.

If the partial sums settle at a particular number as \( n \) becomes very large, then we say the series converges. However, in our case, these partial sums do not settle down. They might grow without bound, oscillate, or behave unpredictably. Such behavior means the series diverges. Thus, by analyzing the sequence of partial sums, we identify whether a series converges or diverges.
Convergence
Convergence is a crucial concept in the world of series. It refers to the condition where the sequence of partial sums of a series approaches a finite limit. In simpler terms, it means that as we keep adding more terms, the sum gets closer and closer to a specific number.

For a series \( \sum_{k=1}^{\infty} a_{k} \), convergence means the sequence \( S_n = a_1 + a_2 + \ldots + a_n \) approaches a finite value as \( n \) becomes large. The opposite, when this doesn't happen, is what we call divergence. A divergent series has partial sums that either grow too large, diminish beyond bound, or oscillate eternally. When the sequence does neither, convergence is assured, and we can comfortably find and express the sum as a real number.
Multiplication by a Constant
A fascinating property of series is what happens when we multiply each term by a constant. If we have an original series \( \sum_{k=1}^{\infty} a_{k} \), and it diverges, then the series \( \sum_{k=1}^{\infty} c a_{k} \) should also diverge when \( c eq 0 \).

To understand why, consider the sequence of partial sums. For the original series, they are \( S_n = a_1 + a_2 + \ldots + a_n \). For the modified series, they become \( T_n = c(a_1 + a_2 + \ldots + a_n) = c S_n \). Thus, the new partial sums are simply the old ones multiplied by the constant \( c \).

The characteristic of divergence is preserved through this multiplication. If \( S_n \) diverges 鈥� going to infinity, negative infinity, or oscillating 鈥� the product \( cS_n \) cannot suddenly become well-behaved or converge to a finite number. Hence, multiplying by a non-zero constant does not change the fact that the series diverges, emphasizing the robust behavior of divergence.

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Most popular questions from this chapter

One can sometimes find a Maclaurin series by the method of equating coefficients. For example, let $$ \tan x=\frac{\sin x}{\cos x}=a_{0}+a_{1} x+a_{2} x^{2}+\cdots $$ Then multiply by \(\cos x\) and replace \(\sin x\) and \(\cos x\) by their series to obtain $$ \begin{aligned} x-\frac{x^{3}}{6}+\cdots &=\left(a_{0}+a_{1} x+a_{2} x^{2}+\cdots\right)\left(1-\frac{x^{2}}{2}+\cdots\right) \\ &=a_{0}+a_{1} x+\left(a_{2}-\frac{a_{0}}{2}\right) x^{2}+\left(a_{3}-\frac{a_{1}}{2}\right) x^{3}+\cdots \end{aligned} $$ Thus, $$ a_{0}=0, \quad a_{1}=1, \quad a_{2}-\frac{a_{0}}{2}=0, \quad a_{3}-\frac{a_{1}}{2}=-\frac{1}{6}, \quad \ldots $$ so $$ a_{0}=0, \quad a_{1}=1, \quad a_{2}=0, \quad a_{3}=\frac{1}{3}, \quad \ldots $$ and therefore $$ \tan x=0+x+0+\frac{1}{3} x^{3}+\cdots $$ which agrees with Problem 1. Use this method to find the terms through \(x^{4}\) in the series for \(\sec x\).

In Problems 9-28, find the convergence set for the given power series. Hint: First find a formula for the nth term; then use the Absolute Ratio Test. $$ \frac{x-2}{1^{2}}+\frac{(x-2)^{2}}{2^{2}}+\frac{(x-2)^{3}}{3^{2}}+\frac{(x-2)^{4}}{4^{2}}+\cdots $$

In Problems 1-20, an explicit formula for \(a_{n}\) is given. Write the first five terms of \(\left\\{a_{n}\right\\}\), determine whether the sequence converges or diverges, and, if it converges, find \(\lim _{n \rightarrow \infty} a_{n}\) \(a_{n}=\frac{n \cos (n \pi)}{2 n-1}\)

A famous sequence \(\left\\{f_{n}\right\\}\), called the Fibonacci Sequence after Leonardo Fibonacci, who introduced it around A.D. 1200 , is defined by the recursion formula $$ f_{1}=f_{2}=1, \quad f_{n+2}=f_{n+1}+f_{n} $$ (a) Find \(f_{3}\) through \(f_{10}\) - (b) Let \(\phi=\frac{1}{2}(1+\sqrt{5}) \approx 1.618034\). The Greeks called this number the golden ratio, claiming that a rectangle whose dimensions were in this ratio was "perfect." It can be shown that $$ \begin{aligned} f_{n} &=\frac{1}{\sqrt{5}}\left[\left(\frac{1+\sqrt{5}}{2}\right)^{n}-\left(\frac{1-\sqrt{5}}{2}\right)^{n}\right] \\\ &=\frac{1}{\sqrt{5}}\left[\phi^{n}-(-1)^{n} \phi^{-n}\right] \end{aligned} $$ Check that this gives the right result for \(n=1\) and \(n=2\). The general result can be proved by induction (it is a nice challenge). More in line with this section, use this explicit formula to prove that \(\lim _{n \rightarrow \infty} f_{n+1} / f_{n}=\phi\). (c) Using the limit just proved, show that \(\phi\) satisfies the equation \(x^{2}-x-1=0\). Then, in another interesting twist, use the Quadratic Formula to show that the two roots of this equation are \(\phi\) and \(-1 / \phi\), two numbers that occur in the explicit formula for \(f_{n}\).

In Problems 1-20, an explicit formula for \(a_{n}\) is given. Write the first five terms of \(\left\\{a_{n}\right\\}\), determine whether the sequence converges or diverges, and, if it converges, find \(\lim _{n \rightarrow \infty} a_{n}\) \(a_{n}=\frac{\sqrt{3 n^{2}+2}}{2 n+1}\)
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