Question

Write out the sum of the first 5 terms of the given power series. $$\sum_{n=1}^{\infty} \frac{1}{n^{2}} x^{n}$$

Short answer

The sum is \( x + \frac{1}{4}x^2 + \frac{1}{9}x^3 + \frac{1}{16}x^4 + \frac{1}{25}x^5 \).

Step-by-step solution

Identify the Pattern of Terms

The given power series is \( \sum_{n=1}^{\infty} \frac{1}{n^2} x^n \). Each term in this series is of the form \( \frac{1}{n^2} x^n \) where \( n \) is an integer starting from 1.

Write Out the First 5 Terms

To find the sum of the first 5 terms, we substitute \( n = 1, 2, 3, 4, \) and \( 5 \) into the series:1. For \( n = 1 \), the term is \( \frac{1}{1^2} x^1 = x \).2. For \( n = 2 \), the term is \( \frac{1}{2^2} x^2 = \frac{1}{4} x^2 \).3. For \( n = 3 \), the term is \( \frac{1}{3^2} x^3 = \frac{1}{9} x^3 \).4. For \( n = 4 \), the term is \( \frac{1}{4^2} x^4 = \frac{1}{16} x^4 \).5. For \( n = 5 \), the term is \( \frac{1}{5^2} x^5 = \frac{1}{25} x^5 \).

Sum the Terms

Add the terms obtained in Step 2:\[ x + \frac{1}{4}x^2 + \frac{1}{9}x^3 + \frac{1}{16}x^4 + \frac{1}{25}x^5. \]

Key concepts

Series Expansion

A power series is a way to express functions as a sum of terms calculated from the powers of a variable and corresponding coefficients. When we talk about series expansion, we are looking at how a function can be represented in an infinite series form. This expansion allows us to approximate complex functions using simpler polynomial-like expressions.

In the case of the exercise, the given series expansion follows the format:

• Term = \( \frac{1}{n^2} x^n \)
• Each term is a fraction where the numerator is 1 and the denominator is a perfect square depending on \( n \).
• \( x^n \) indicates that each subsequent term increases in power of \( x \).

For practical purposes, especially in homework or when applied in scientific calculations, we often use the sum of the first few terms. This helps us gain a reasonable approximation of the function's behavior.

By calculating only the first five terms, like in our solution, you can approximate the series without dealing with the full complexity of an infinite sequence.

Convergence of Series

One significant aspect of a power series is whether it converges or diverges. Convergence of a series means that as more terms are added, the series approaches a finite limit. For a series to converge, the terms need to become smaller and tend toward zero as they approach infinity.

The series given in the exercise is \[\sum_{n=1}^{\infty} \frac{1}{n^2} x^n \].

• Notice that \( \frac{1}{n^2} \) decreases as \( n \) becomes larger, suggesting convergence.
• The variable \( x \) affects the convergence; if \( |x| < 1 \), the series tends to converge due to the decreasing power of \( x^n \).
• Conversely, if \( |x| \geq 1 \), the series might diverge depending on how rapidly the terms decrease.

Understanding convergence is crucial because it assures that our series expansion provides meaningful and usable results. When working with series expansion in practical applications, confirming convergence is essential to ensure that results are accurate.

Mathematical Notation

Mathematical notation is the symbolic representation of numbers, operations, and relationships in mathematics. It plays an essential role in expressing ideas clearly and efficiently, especially in complex concepts like series. In dealing with power series, specific notations are frequently used, which must be understood for a clear comprehension.

Key notations in the exercise include:

• \( \sum \): This symbol represents the summation of terms, starting from an initial index up to infinity.
• \( n \): The index of summation, indicating the sequence of terms in the series.
• \( x^n \): Represents the power of \( x \), showing the increasing order in each term.
• \( \frac{1}{n^2} \): Represents the coefficients of each term, illustrating the inverse square pattern.

Understanding mathematical notation is fundamental because it allows us to decode expressions quickly, enabling us to grasp the underlying principles they describe. By mastering these symbols, students can effortlessly navigate through complex equations and series manipulations, enhancing their problem-solving abilities.