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Magazine Chapter 5: Problem 17
Express in summation notation. \(1-\frac{x^{2}}{2}+\frac{x^{4}}{4}-\frac{x^{6}}{6}+\cdots+(-1)^{n} \frac{x^{2 n}}{2 n}\)
Short Answer
Expert verified
\(\sum_{k=0}^{n} (-1)^{k} \frac{x^{2k}}{2k}\).
Step by step solution
01
Identify the Pattern
Recognize the terms in the given expression. We have a sequence with terms: 1, \(-\frac{x^{2}}{2}\), \(\frac{x^{4}}{4}\), \(-\frac{x^{6}}{6}\), \(\cdots\), \((-1)^{n}\frac{x^{2n}}{2n}\). Each term consists of a combination of an alternating sign, powers of \(x\), and division by an even number.
02
Formulate the General Term
Each term of the sequence can be expressed in a common form: \((-1)^{k}\frac{x^{2k}}{2k}\). Here, \(k\) represents an integer starting from 0.The signs alternate due to the factor \((-1)^{k}\), raising \(x\) to increasingly even powers, \(2k\), and dividing by \(2k\).
03
Construct the Summation Expression
To express the series in summation notation, sum the general term over \(k\) from 0 to \(n\):\[\sum_{k=0}^{n} (-1)^{k} \frac{x^{2k}}{2k}\].
04
Verify the Summation
Check that the terms generated from the summation expression match those in the original series. When \(k=0\), the term is \(1\), as given. When \(k=1\), the term is \(-\frac{x^2}{2}\), and so on, confirming the pattern of alternating signs and decreasing denominator matches the original expression.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Alternating Series
An alternating series is a sequence where the signs of the terms alternate between positive and negative. This particular series:
- Starts with a positive number.
- Alternates to a negative term.
- Continues this pattern, progressively alternating signs.
- When \k\ is even, the term is positive.
- When \k\ is odd, the term is negative.
Powers of x
Powers of \(x\) refer to exponential terms where \(x\) is raised to a particular power. In this series, each term contains \(x^{2k}\). Key points here include:
- The exponent of \(x\) increases sequentially in even numbers, such as \{0, 2, 4, \ldots, 2n\}.
- This means in each successive term, the power of \(x\) increases by 2.
General Term
The concept of a general term is crucial for expressing sequences succinctly. Here, it represents a formula that describes every term in the series. For our series the general term is \((-1)^{k} \frac{x^{2k}}{2k}\).
- The alternating sign is given by \((-1)^k\).
- \(x\) is raised to the \(2k\)-th power.
- The whole term is divided by \(2k\).
Series Expression
A series expression is a mathematical notation of a sequence expressed in summation form. It efficiently conveys the sum of a sequence's terms as shown with the notation: \[ \sum_{k=0}^{n} (-1)^{k} \frac{x^{2k}}{2k} \]
- The series starts at \(k = 0\) and ends at \(k = n\).
- The summation captures the pattern of terms derived from the general term.
Mathematical Notation
Mathematical notation is a universal language used to convey mathematical ideas clearly and concisely. This problem utilizes key notational tools such as:
- Use of the summation symbol \(\sum\) to denote the sum of terms.
- Elements like \( (-1)^k \) for alternating signs.
- Subscripts and superscripts, such as \(k = 0\) and \(x^{2k}\), to indicate limits and powers.
