Question

Suppose that the two power series \(\sum_{n=0}^{\infty} c_{n} x^{n}\) and \(\sum_{n=0}^{\infty} d_{n} x^{n}\) converge to the functions \(f\) and \(g\), respectively, on a common interval \(l\). i. The power series \(\sum_{n=0}^{\infty}\left(c_{n} x^{n} \pm d_{n} x^{n}\right)\) converges to \(f \pm g\) on \(l\). ii. For any integer \(m \geq 0\) and any real number \(b\), the power series \(\sum_{n=0}^{\infty} b x^{m} c_{n} x^{n}\) converges to \(b x^{m} f(x)\) on \(I\) iii. For any integer \(m \geq 0\) and any real number \(b\), the series \(\sum_{n=0}^{\infty} c_{n}\left(b x^{m}\right)^{n}\) converges to \(f\left(b x^{m}\right)\) for all \(x\) such that \(b x^{m}\) is in \(I\).

Short answer

The series converge as described: (i) \( f \pm g \), (ii) \( b x^m f(x) \), and (iii) \( f(bx^m) \).

Step-by-step solution

Understanding Power Series Convergence

We begin by recognizing that each power series represents a function: \( f(x) = \sum_{n=0}^{\infty} c_{n} x^{n} \) and \( g(x) = \sum_{n=0}^{\infty} d_{n} x^{n} \). These series converge to their respective functions on a common interval \( l \). This means that for any value of \( x \) within \( l \), the series sum equals their function value.

Combining Series

We need to consider the sum and difference of the two given power series. Specifically, the series \( \sum_{n=0}^{\infty} (c_{n} x^{n} \pm d_{n} x^{n}) \). The sum of these series can be expressed as: \( \sum_{n=0}^{\infty} c_{n} x^{n} + \sum_{n=0}^{\infty} d_{n} x^{n} = f(x) + g(x) \) and the difference as \( f(x) - g(x) \). Therefore, both converge to their respective functions \( f \pm g \) over interval \( l \).

Scaling the Series

Consider a scaled version of the first power series \( \sum_{n=0}^{\infty} b x^{m} c_{n} x^{n} \). This can be rewritten as \( bx^m \sum_{n=0}^{\infty} c_n x^n \). Since the original series converges to \( f(x) \), scaling it by \( b x^m \) results in \( b x^m f(x) \), which converges over the same interval \( I \).

Changing the Variable

We examine the transformation of the series through substitution: \( \sum_{n=0}^{\infty} c_{n} (b x^{m})^n \), which is equivalent to \( \sum_{n=0}^{\infty} c_{n} (b^n x^{mn}) \). If \( b^nx^m \) lies within the convergence interval \( I \), this series converges to \( f(bx^m) \). This follows from substituting \( x \) with \( bx^m \) in the original series for \( f(x) \).

Key concepts

Convergence of Power Series

Understanding the concept of convergence is key when working with power series. A power series, like the two we have here \(\sum_{n=0}^{\infty} c_{n} x^{n}\) and \(\sum_{n=0}^{\infty} d_{n} x^{n}\), converges if the sum of the series approaches a specific value as more terms are added.
This specific value is the function the series represents. In practical terms, for any value of \(x\) within the interval \(l\), both series will "converge" or amount to the exact values of the functions \(f(x)\) and \(g(x)\), respectively.
This interval \(l\) is where all the magic happens, ensuring the series results match their respective function outputs.

Interval of Convergence

The interval, denoted as \(l\), is a range of values on which our power series converge to their respective functions \(f(x)\) or \(g(x)\).
It's like a safe zone! Inside this interval, you can substitute any value of \(x\) and be sure the series will neatly wrap up to the desired function value.
However, outside this zone, the series might not behave as predictably. This interval is vital as it defines the scope of the series' applicability.
For practical exercises, always determine and work within this interval to ensure the validity of convergence.

Function Transformation in Series

Function transformation in a power series lets you explore infinite realms of mathematical possibility.
When we talk about function transformations, we mostly deal with changes in scale or structure within our series. For example, consider scaling the series: \(\sum_{n=0}^{\infty} b x^{m} c_{n} x^{n}\).

• This transformation introduces a factor \(b x^m\) that modifies the original series \(\sum_{n=0}^{\infty} c_n x^n\).

The process results in the function \(b x^m f(x)\) over the same interval \(I\), essentially scaling \(f(x)\) by \(b x^m\).
This manipulation allows you to create new functions from existing series, always visualized within the correct convergence interval.

Sum and Difference of Series

When dealing with the sum and difference of power series, it's as straightforward as combining like terms in basic algebra.
Taking sums and differences, like in our series \(\sum_{n=0}^{\infty} (c_{n} x^{n} \pm d_{n} x^{n})\), the combination rule applies.

• The sum \(\sum_{n=0}^{\infty} c_{n} x^{n} + \sum_{n=0}^{\infty} d_{n} x^{n}\) results in the new function \(f(x) + g(x)\).
• The difference, quite similarly, results in \(f(x) - g(x)\).

This combining ensures that both resulting functions converge on the specified interval \(l\).
Understanding this concept helps in manipulating and combining series in a mathematical setup, broadening the scope of function approximation and preparation for higher-level calculus work.