Problem 14
Question
Derive the indicated result by appealing to the geometric series. $$\sum_{i=1}^{2}(-1)^{k} x^{2 k}-\frac{1}{1-x^{2}} \cdot|x|<1$$
Step-by-Step Solution
Verified Answer
The sum \(\sum_{i=1}^{2}(-1)^k x^{2k}\) becomes a geometric series with a = \(x^2\), r= -\(x^2\) and n=2 which can be reformulated to -\(\frac{1}{x^2}\)from the geometric series. Comparing this result to the series derived for \(\frac{1}{1-x^2}\), we can indeed see that these results are equivalent.
1Step 1: Understanding the geometric series and its formula
A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number. The sum of the geometric series is given by the formula \(S = a \cdot \frac{1-r^n}{1-r}\), where a is the first term, r is the common ratio and n is the number of terms.
2Step 2: Identifying the elements of our geometric series
From the series, we can identify a= \(x^2\), r= -\(x^2\), and n=1. After the summation, the terms can be rearranged.
3Step 3: Calculating the power series from the rearranged equation
Now, the power series of \(-\frac{1}{x^2}\) which is the rearranged equation \(\frac{1}{1-(-x^2)}\) can be derived to be \(\sum_{i=1}^{2}(-1)^k x^{2k}\) by using the geometric series formula.
4Step 4: Final computation
Comparing the derived sum and the equation under the summation reveals they are equivalent, as needed.\(-\frac{1}{x^2} = \sum_{i=1}^{2}(-1)^k x^{2k}\) is the derived result.
Key Concepts
SequencePower SeriesSum of Series
Sequence
In mathematics, a sequence is a collection of numbers in an ordered list. Each number in a sequence is called a "term." Sequences can be finite, having a finite number of terms, or infinite, which go on indefinitely. Sequences are foundational to understanding many advanced mathematical concepts, particularly those involving series and calculus.
To comprehend sequences more fully, it's essential to recognize that they may follow a particular rule or pattern. For example, in an arithmetic sequence, each term is obtained by adding a fixed number to the previous term, whereas in a geometric sequence, each term is derived by multiplying the previous term by a fixed number.
To comprehend sequences more fully, it's essential to recognize that they may follow a particular rule or pattern. For example, in an arithmetic sequence, each term is obtained by adding a fixed number to the previous term, whereas in a geometric sequence, each term is derived by multiplying the previous term by a fixed number.
- An example of an arithmetic sequence is: 2, 4, 6, 8, ... where the fixed number added is 2.
- A geometric sequence example is: 3, 6, 12, 24, ... with a fixed multiplier of 2.
Power Series
A power series is an infinite series of the form \(\sum_{i=0}^{\infty} a_i (x-c)^i\), where \(a_i\) represents the coefficients, and \(c\) is a constant termed the center of the series.
In essence, power series can be thought of as extended polynomials of infinite degree, and they play an essential role in calculus and analytic functions.
A simple and well-known power series is the geometric series, which takes the form \(\sum_{i=0}^{\infty} r^i\) where the series converges to \(\frac{1}{1-r}\) if \(|r| < 1\). This is central to solving problems involving the convergence of series.
In the original solution, the concept of the power series ties in directly with converting the given sequence into a more manageable and comparable form against the series' standard formula. By using the properties of a geometric series, we identify \(a = x^2\) and \(r = -x^2\), providing a basis to equate and rearrange the series.
In essence, power series can be thought of as extended polynomials of infinite degree, and they play an essential role in calculus and analytic functions.
A simple and well-known power series is the geometric series, which takes the form \(\sum_{i=0}^{\infty} r^i\) where the series converges to \(\frac{1}{1-r}\) if \(|r| < 1\). This is central to solving problems involving the convergence of series.
In the original solution, the concept of the power series ties in directly with converting the given sequence into a more manageable and comparable form against the series' standard formula. By using the properties of a geometric series, we identify \(a = x^2\) and \(r = -x^2\), providing a basis to equate and rearrange the series.
Sum of Series
The sum of a series is essentially the addition of the terms of a sequence. Calculating this sum can demonstrate convergence—a key property in mathematical analysis where the series approaches a specific value.
For a geometric series, the sum can be computed using the formula \(S = a \cdot \frac{1-r^n}{1-r}\), provided the absolute value of the common ratio \(r\) is less than one, ensuring the series converges.
The original exercise poses a series \(\sum_{i=1}^{2}(-1)^k x^{2k}\), and by recognizing it as a geometric series, we can apply the geometric series sum formula.
For a geometric series, the sum can be computed using the formula \(S = a \cdot \frac{1-r^n}{1-r}\), provided the absolute value of the common ratio \(r\) is less than one, ensuring the series converges.
The original exercise poses a series \(\sum_{i=1}^{2}(-1)^k x^{2k}\), and by recognizing it as a geometric series, we can apply the geometric series sum formula.
- Here, the first term (a) would be \(x^2\).
- The common ratio (r) is \(-x^2\).
- As this is a finite series with terms only up to 2, n is set as 1, simplifying our computation.
Other exercises in this chapter
Problem 14
Test these series for (a) absolute convergence, (b) conditional convergence. \(\sum(-1)^{k} \frac{k}{k^{2}+1}\).
View solution Problem 14
Determine whether the series converges or diverges. $$\sum \frac{k^{2}}{e^{k}}$$
View solution Problem 14
Find the interval of convergence. $$\sum \frac{1}{\ln k} x^{k}$$
View solution Problem 14
Determine the \(n\) th Taylor polynomial \(P_{n}\) for the function \(f\). $$f(x)=\ln (1-x)$$
View solution