10.8 Testing Convergence at Endpoints of Intervals of Convergence

• What is the general form of a power series?

  $\sum_{n = 0}^{\infty} a_{n}(x - c)^{n}$
• A power series is centered at the value c
• The power series $\sum_{n = 0}^{\infty} x^{n}$ is centered at $0$.
• What is the interval of convergence for a power series?

  Values of x for convergence
• The radius of convergence determines the interval of convergence
• The ratio test states that a series converges if \lim_{n \to \infty} \left| \frac{a_{n + 1}}{a_{n}} \right| |x - c| < 1</latex>.
• Steps to find the interval of convergence for a power series

  1️⃣ Apply the ratio or root test to find R

  2️⃣ Determine the interval (c - R, c + R)

  3️⃣ Test the endpoints c - R and c + R
• What does the alternating series test check for?

  Convergence of alternating series
• The harmonic series $\sum_{n = 1}^{\infty} \frac{1}{n}$ diverges.
• The interval of convergence for $\sum_{n = 1}^{\infty} \frac{x^{n}}{n}$ is [ - 1, 1)
• What is the purpose of testing endpoints when finding the interval of convergence?

  To determine convergence at boundaries
• Match the convergence test with its description:

  Comparison Test ↔️ Compares with known series

  Limit Comparison Test ↔️ Uses the limit of the ratio

  Alternating Series Test ↔️ Checks alternating signs and terms

  Divergence Test ↔️ Checks if terms approach zero
• The divergence test states that if the terms of a series do not approach zero, the series diverges.
• The comparison test requires terms to be positive and compared with a series of known convergence.
• The limit comparison test evaluates $\lim_{n \to \infty} \frac{a_{n}}{b_{n}} =$$L$, where 0 < L < \infty</latex> implies convergence of $\sum a_{n}$ if $\sum b_{n}$ converges.
• Steps to verify convergence using the alternating series test

  1️⃣ Verify alternating signs

  2️⃣ Show $\lim_{n \to \infty} |a_{n}| =$$0$

  3️⃣ Confirm $|a_{n + 1}| \leq |a_{n}|$
• If \lim_{n \to \infty} a_{n} \neq 0</latex>, the divergence test implies the series diverges.
• What is the term 'c' called in a power series?

  Center
• The geometric series $\sum_{n = 0}^{\infty} x^{n}$ is centered at $c =$$0$.
• The interval of convergence for a power series is the set of $x$ values for which the series converges.
• Which tests are used to find the radius of convergence?

  Ratio or root test
• Steps to determine the interval of convergence

  1️⃣ Find the radius of convergence $R$

  2️⃣ Identify the interval $(c - R, c + R)$

  3️⃣ Test the endpoints $c - R$ and $c +$$R$
• The series $\sum_{n = 1}^{\infty} \frac{1}{n}$ diverges at $x =$$1$.
• When testing endpoints for convergence, the divergence test checks if the terms approach zero.
• The comparison test can be used to show that \sum \frac{1}{n^{2} + 1} converges by comparing it to \sum \frac{1}{n^{2}}</latex>.
• The alternating series test requires terms to alternate in sign.
• What is the primary condition for using the Comparison Test?

  Terms are positive
• The series \sum \frac{1}{n^{2} + 1} converges by comparing it to the series $\sum \frac{1}{n^{2}}$, which is a known p-series
• The Limit Comparison Test requires the limit of the ratio $\frac{a_{n}}{b_{n}}$ to be a finite, non-zero number for $\sum a_{n}$ and $\sum b_{n}$ to either both converge or both diverge.
• Steps to verify the conditions for the Alternating Series Test

  1️⃣ Verify alternating signs

  2️⃣ Ensure $\lim_{n \to \infty} |a_{n}| =$$0$

  3️⃣ Confirm $|a_{n + 1}| \leq |a_{n}|$
• When does the Divergence Test indicate that a series diverges?

  \lim_{n \to \infty} a_{n} \neq 0</latex>
• The Limit Comparison Test is particularly effective when the series terms $a_{n}$ resemble the terms of a known series, such as a p-series
• Both $\sum a_{n}$ and $\sum b_{n}$ in the Limit Comparison Test must have positive terms.
• Steps to apply the Limit Comparison Test

  1️⃣ Choose a comparison series $\sum b_{n}$

  2️⃣ Calculate the limit $\lim_{n \to \infty} \frac{a_{n}}{b_{n}} =$$L$

  3️⃣ Determine convergence based on the value of $L$
• Determine if the series $\sum \frac{3n + 5}{n^{2} - 1}$ converges or diverges using the Limit Comparison Test.


  Diverges
• The series $\sum \frac{3n + 5}{n^{2} - 1}$ diverges because the limit \lim_{n \to \infty} \frac{3n^{2} + 5n}{n^{2} - 1}</latex> equals 3
• An alternating series has terms that alternate in sign.
• What are the conditions for the Alternating Series Test to determine convergence?

  Decreasing terms and zero limit
• The alternating series $\sum \frac{( - 1)^{n - 1}}{n}$ converges because the terms are decreasing and $\lim_{n \to \infty} \frac{1}{n} =$$0$, which satisfies the Alternating Series Test
• If an alternating series satisfies the conditions of the Alternating Series Test, it always converges.