10.11 Radius and Interval of Convergence of Power Series

• What is the general form of a power series?

  $\sum_{n = 0}^{\infty} c_{n}(x - a)^{n}$
• The center of a power series is denoted by the variable a
• What is the radius of convergence?

  The distance around the center where the series converges
• The radius of convergence defines the range of x</latex> values for which a power series is guaranteed to converge.
• The radius of convergence is denoted by the variable R
• If the radius of convergence $R =$$1$ and the center is a = 2</latex>, what is the range of $x$ values for convergence?


  $1 \le x \le 3$
• Match the aspect with its definition:

  Radius of Convergence ↔️ The distance from the center where the series converges

  Center ↔️ The point about which the power series is defined

  Convergence ↔️ The property of the series to approach a finite value
• The interval of convergence extends beyond the radius of convergence to include the endpoints
• The interval of convergence can include $(a - R, a + R)$, $[a - R, a + R]$, or $(a - R, a + R]$ depending on the convergence at the endpoints.
• What is the condition for convergence in the ratio test?

  \lim_{n \to \infty} \left| \frac{c_{n + 1} (x - a)^{n + 1}}{c_{n} (x - a)^{n}} \right| < 1</latex>
• The power series $\sum_{n = 0}^{\infty} \frac{(x - 3)^{n}}{n!}$ has a radius of convergence \infty
• What does the radius of convergence define?

  The range of $x$ values for guaranteed convergence
• What are the coefficients in a power series called?

  Coefficients
• The center of a power series is denoted by the variable a
• The radius of convergence is essential for understanding the behavior of a power series.
• What does the radius of convergence define in a power series?

  The distance from the center where the series converges
• The interval of convergence includes the endpoints a - R and $a +$$R$
• If both endpoints of the interval of convergence converge, the interval is expressed as $[a - R, a + R]$.
• What is the condition for convergence if both endpoints diverge in the interval of convergence?

  $(a - R, a + R)$
• The ratio test states that a series converges if \lim_{n \to \infty} \left| \frac{c_{n + 1} (x - a)^{n + 1}}{c_{n} (x - a)^{n}} \right| < 1
• What is the radius of convergence for the power series \sum_{n = 0}^{\infty} \frac{(x - 3)^{n}}{n!}</latex>?

  $R =$$\infty$
• Steps to find the radius of convergence using the ratio test

  1️⃣ Apply the ratio test to the power series

  2️⃣ Simplify the expression

  3️⃣ Solve for $|x - a|$

  4️⃣ Set $|x - a| < R$ to determine the convergence range
• To find the interval of convergence, after determining the radius of convergence $R$, you must test the endpoints
• The series $\sum_{n = 1}^{\infty} \frac{( - 1)^{n}}{n}$ converges by the Alternating Series Test.
• What does the series $\sum_{n = 1}^{\infty} \frac{1}{n}$ diverge to?


  Infinity
• Match the endpoint convergence with the resulting series behavior:

  $x =$$a - R$ ↔️ Converges

  $x =$$a +$$R$ ↔️ Diverges
• What must you test after determining the radius of convergence of a power series to find the interval of convergence?

  The endpoints
• To find the interval of convergence, you must substitute $x =$$a - R$ and $x =$$a +$$R$ into the power series
• Steps to find the interval of convergence after determining the radius of convergence.

  1️⃣ Substitute x = a - R</latex> and $x =$$a +$$R$ into the power series

  2️⃣ Evaluate each resulting series

  3️⃣ Determine whether each series converges or diverges

  4️⃣ Combine the convergence results with the radius of convergence
• The radius of convergence determines whether a power series converges or diverges for all $x$ values.


  False
• What are the values of $a$ and $R$ for the power series \sum_{n = 1}^{\infty} \frac{(x - 2)^{n}}{n}</latex>?


  $a =$$2, R =$$1$
• For $x =$$1$, the series $\sum_{n = 1}^{\infty} \frac{( - 1)^{n}}{n}$ converges by the Alternating Series Test
• The interval of convergence for the power series $\sum_{n = 1}^{\infty} \frac{(x - 2)^{n}}{n}$ is $[1, 3)$.
• What are two common types of series used to approximate functions?

  Taylor and Maclaurin
• The Taylor series expands a function $f(x)$ about a point $x =$$a$ using its derivatives
• What is the value of $a$ in the Maclaurin series?


  a = 0</latex>
• Steps to approximate $f(x) =$$e^{x}$ using a Maclaurin series up to the third term.


  1️⃣ Calculate $f(0)$

  2️⃣ Find the first, second, and third derivatives of $f(x)$

  3️⃣ Evaluate the derivatives at $x =$$0$

  4️⃣ Substitute the values into the Maclaurin series formula
• What is the approximation of $e^{x}$ using a Maclaurin series up to the third term?


  $1 +$$x +$$\frac{x^{2}}{2!} +$$\frac{x^{3}}{3!}$
• The Maclaurin series is a special case of the Taylor series with $a =$$0$.
• What is the general form of a power series centered ata</latex>?

  $\sum_{n = 0}^{\infty} c_{n}(x - a)^{n}$