Question

Rewrite the given expression as a sum whose generic term involves \(x^{n} .\) $$ \sum_{n=0}^{\infty} a_{n} x^{n+2} $$

Short answer

Question: Rewrite the given sum as a sum whose generic term involves \(x^n\): \(\sum_{n=0}^{\infty} a_{n} x^{n+2}\) Answer: \(\sum_{n=2}^{\infty} a_{n-2} x^{n}\)

Step-by-step solution

Compare the generic terms

Notice that the given sum is \(\sum_{n=0}^{\infty} a_{n} x^{n+2}\) and we want to rewrite it to be in the form of a sum whose generic term involves \(x^n\). To do this, we just need to replace the \((n+2)\) exponent.

Rewrite the exponent

Replacing \((n+2)\) exponent with \(n\), we get $$ \sum_{n=0}^{\infty} a_{n} x^{n+2} = \sum_{n=2}^{\infty} a_{n-2} x^{n}. $$

Check the solution

Now, the given expression has been rewritten as a sum whose generic term involves \(x^n\).

Key concepts

Sum Notation

Sum notation, or sigma notation, is a concise and powerful way to represent the addition of a sequence of numbers. In the context of power series, sum notation allows us to write a potentially infinite series of terms using the symbol \(\sum\). Understanding sum notation is crucial for students tackling exercises involving series and sequences.

For example, when you see \(\sum_{n=0}^{\infty} a_n x^{n+2}\), this tells you several things. The index of summation, \(n\), starts at 0 and continues indefinitely- this is indicated by the \(\infty\) symbol. Each term of the series is represented by \(a_n x^{n+2}\), where \(a_n\) reflects a coefficient that can change with each \(n\), and \(x\) is raised to the power of \(n+2\). By using sum notation, a potentially endless string of terms can be neatly summarized, which is vital for both neatness and clarity in mathematics.

The sum notation is particularly helpful because it can encapsulate rules about how the series behaves and evolves with each subsequent term, offering a blueprint for the series in a compact form. Students will often encounter this notation when working with power series, where understanding how to interpret and manipulate it becomes critical.

Infinite Series

An infinite series is a sum of terms that goes on indefinitely. Its beauty lies in its ability to represent complex functions and numbers that might be challenging to describe otherwise. Many functions in mathematics can be expanded into their power series form—an infinite series where each term involves a power of \(x\).

In practical exercises, you often see an infinite series written with sum notation, such as \(\sum_{n=0}^{\infty} a_{n} x^{n}\), which represents the series as the sum of \(a_{n} x^{n}\) terms, starting with \(n=0\) and continuing endlessly. It's important to note that despite their 'infinite' label, these series can converge to finite values if the conditions are right—this is a fascinating aspect of calculus and analysis that students dive into when studying convergence tests and series manipulation.

It's essential for students to feel confident in recognizing and working with infinite series because they're foundational in higher mathematics. They are used to approximate functions, solve differential equations, and are integral to the fields of engineering, physics, and beyond. Understanding the representation and properties of infinite series offers students a gateway to mastering complex mathematical concepts.

Exponent Manipulation

Exponent manipulation refers to the methods and rules we apply to simplify or reshape expressions involving exponents. In power series, we frequently need to adjust the exponents to meet a particular form or to facilitate further calculation. This exercise involves a basic but fundamental aspect of exponent manipulation: changing the index of a series to simplify the expression or to fit a desired pattern.

For instance, translating \(\sum_{n=0}^{\infty} a_{n} x^{n+2}\) into \(\sum_{n=2}^{\infty} a_{n-2} x^{n}\) requires a shift in perspective. We're essentially 're-indexing' the series—adjusting the exponent so that the \(x\) term aligns with the standard form \(x^n\) while ensuring the coefficients \(a_n\) correlate correctly with the new index.

This kind of exponent manipulation is fundamental in mathematics, as it allows for the comparison of different series, the determination of convergence, and the application of various series-centered techniques. By mastering exponent manipulation, students can unlock the full potential of power series and harness these tools for various applications, from theoretical proofs to real-world problem-solving scenarios.