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Magazine Chapter 9: Problem 3
Which of the series in Exercises are power series? $$1+x+(x-1)^{2}+(x-2)^{3}+(x-3)^{4}+\cdots$$
Short Answer
Expert verified
The series is not a power series because it does not have a consistent center.
Step by step solution
01
Understand Power Series Definition
A power series is a series of the form \( \sum_{n=0}^{\infty} a_n (x-c)^n \), where \( a_n \) are constants, \( x \) is a variable, and \( c \) is the center of the series. Each term involves powers of \((x-c)\).
02
Identify Pattern of Given Series
Examine the series given: \(1 + x + (x-1)^{2} + (x-2)^{3} + (x-3)^{4} + \cdots\). Notice the pattern of exponents and terms. Each term \((x-k)^k\) increases the power of \(x-k\) by 1 as \(k\) increases by 1.
03
Check the Series against Power Series Form
For a series to be a power series, it must have a consistent center \(c\). Here, each term in the series is of the form \((x-k)^k\) which changes \(k\) for each term, but the center \(c\) is not constant. This is not consistent with the power series form.
04
Conclude if Series is a Power Series
Since a power series requires a consistent center \(c\) and the given series changes the center for each term, it does not fit the definition of a power series.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Series Definition
To grasp what a power series is, it's essential to start with its basic definition. A power series is expressed in this format: \[ \sum_{n=0}^{\infty} a_n (x-c)^n \] Here,
- \( a_n \) represents a sequence of coefficients which are constants specific to each term.
- \( x \) is the variable that all terms in the series depend on.
- \( c \) is known as the center of the series. It is crucial because it influences the formation of each term in the series.
Pattern Recognition
Examining patterns within a sequence is an excellent method for better comprehension. Consider the given series: \[ 1 + x + (x-1)^{2} + (x-2)^{3} + (x-3)^{4} + \cdots \]The pattern can be readily identified by observing the exponents and the terms:
- Each term looks like this: \((x-k)^k\), where \(k\) starts at zero and increases by one with each new term.
- The exponent \(k\) is directly equal to the offset from \(x\), meaning the power applied to the expression is always equal to the number subtracted from \(x\).
Consistent Center
In the realm of power series, the consistency of the center \(c\) is vital. In a power series, \((x-c)^n\), the center \(c\) remains unchanged throughout all terms in the sequence.
- This constancy ensures that while the exponent \(n\) increases sequentially, the expression inside the parenthesis holds steady, revolving only around \(c\).
- For the series provided, as each term changes its base from \(x\) to \(x-1\), \(x-2\), and so on, it suggests a shift in the series' foundation, which violates the fundamental structure of a power series.
Convergence of Series
Understanding series often involves determining if it converges. Convergence refers to whether the sums of the series approach a specific number as more and more terms are added. However, for it to be applicable in a power series context, structure matters greatly.
- In power series terms, convergence only has meaning when the consistent center is established, as the convergence conditions will generally revolve around this center, \(c\).
- Since the highlighted series lacks this regularity, discussing convergence in the traditional sense is problematic.
