10.12 Representing Functions as Power Series

• What is the general form of a power series?

  $\sum_{n = 0}^{\infty} c_{n} (x - a)^{n}$
• The radius of convergence defines the distance from the center within which a power series converges
• If the radius of convergence of a power series is 0, the series diverges for all $x \neq a$.
• The exponential function $e^{x}$ has a radius of convergence equal to \infty
• What is the interval of convergence for the sine function $\sin x$?


  $( - \infty, \infty )$
• The natural logarithm $\ln(1 + x)$ has an interval of convergence ( - 1, 1 ]
• What is the power series representation of the geometric series $\frac{1}{1 - x}$?


  $\sum_{n = 0}^{\infty} x^{n}$
• Common power series representations are essential tools for approximating functions and solving differential equations.
• Within what type of interval can common functions be represented as power series?

  Interval of convergence
• The power series for $\sin x$ converges for all real numbers.
• Match the function with its power series and radius of convergence:

  $\ln(1 + x)$ ↔️ $\sum_{n = 1}^{\infty} \frac{( - 1)^{n + 1} x^{n}}{n}$, 1

  $\frac{1}{1 - x}$ ↔️ $\sum_{n = 0}^{\infty} x^{n}$, 1
• What are power series commonly used for in calculus?

  Approximating functions
• The power series for $\sin x$ has an infinite radius of convergence.
• Where is the power series representation for common functions centered?

  0
• A geometric series converges if $|r| < 1$, where $r$ is the common ratio
• A geometric series with a common ratio of $|r| > 1$ will converge.


  False
• Steps to represent a function as a geometric series:

  1️⃣ Identify the common ratio

  2️⃣ Rewrite the function in the form $\frac{a}{1 - r}$

  3️⃣ Express the function as $\sum_{n = 0}^{\infty} ar^{n}$

  4️⃣ Determine the interval of convergence
• What is the geometric series representation of $\frac{1}{1 - x}$?


  $\sum_{n = 0}^{\infty} x^{n}$
• What is the formula for the sum of a convergent geometric series?

  $\frac{a}{1 - r}$
• A geometric series converges if the absolute value of the common ratio is less than 1.
• What does term-by-term differentiation allow in power series?

  Differentiating each term separately
• Term-by-term differentiation and integration change the radius of convergence of a power series.

  False
• What are the three algebraic manipulations used to convert functions into power series form?

  Substitution, factoring, partial fractions
• Replacing $x$ with $kx$ in a power series is an example of substitution
• What type of algebraic manipulation involves extracting common factors to simplify expressions?

  Factoring
• Algebraic manipulations such as substitution are used to match known power series
• Partial fraction decomposition involves splitting rational functions into simpler fractions
• Why are algebraic manipulations essential for converting functions into power series form?

  To simplify complex expressions
• Substitution in algebraic manipulations involves replacing variables to match known power series.
• Factoring is used to extract common factors in expressions.
• What is the ultimate goal of mastering algebraic manipulation techniques?

  To represent functions as power series
• A power series is an expression of the form $\sum_{n = 0}^{\infty} c_{n} (x - a)^{n}$, where \( c_n \) are coefficients
• What defines the distance from the center within which a power series converges?

  Radius of convergence
• A power series diverges if its limit approaches infinity or oscillates.
• The power series $\sum_{n = 0}^{\infty} x^{n}$ converges for $|x| < 1$, with a radius of convergence 1
• What two types of series are closely related to power series?

  Taylor and Maclaurin series
• Match the function with its power series representation:

  e^{x} ↔️ $\sum_{n = 0}^{\infty} \frac{x^{n}}{n!}$

  \sin x ↔️ $\sum_{n = 0}^{\infty} \frac{( - 1)^{n} x^{2n + 1}}{(2n + 1)!}$

  \cos x ↔️ $\sum_{n = 0}^{\infty} \frac{( - 1)^{n} x^{2n}}{(2n)!}$
• Common power series representations are essential tools for approximating functions and solving differential equations.
• A geometric series converges if the absolute value of its common ratio is less than 1
• Steps to represent a function as a geometric series:

  1️⃣ Identify the common ratio.

  2️⃣ Rewrite the function in the form $\frac{a}{1 - r}$.

  3️⃣ Express the function as $\sum_{n = 0}^{\infty} ar^{n}$.

  4️⃣ Determine the interval of convergence.