10.15 Manipulating Series

• The Root Test is effective for series with exponential terms.

  False
• The Limit Comparison Test requires comparing to a series with known behavior
• Steps to present practice problems and examples effectively

  1️⃣ Use a table to organize problems, solutions, and explanations

  2️⃣ Bold the key steps in the solutions

  3️⃣ Incorporate LaTeX for mathematical expressions

  4️⃣ Refer back to relevant concepts
• What is the sum of the geometric series \sum_{n = 1}^{\infty} \frac{1}{2^{n}}</latex>?

  1
• A p-series converges if $p > 1$.
• Match the series type with its key feature:

  Geometric Series ↔️ Constant ratio between terms

  p-series ↔️ Converges if $p > 1$

  Telescoping Series ↔️ Partial sums simplify

  Alternating Series ↔️ Terms alternate signs
• The addition of two convergent series results in a new series that also converges
• What is the general form of a geometric series?

  $a +$$ar +$$ar^{2} +$$\dots$
• In a telescoping series, consecutive terms cancel out.
• What criterion is used to test for convergence of an alternating series?

  Leibniz's criterion
• A geometric series is characterized by a constant ratio
• The p-series \sum_{n = 1}^{\infty} \frac{1}{n^{3}}</latex> converges.
• What value does the telescoping series $\sum_{n = 1}^{\infty} \left( \frac{1}{n} - \frac{1}{n + 1} \right)$ converge to?


  1
• An alternating series converges if its terms decrease and approach zero
• Convergent series can be added or subtracted.
• What does it mean for a series to converge?

  Partial sums approach a limit
• A series diverges if its partial sums do not approach a finite limit
• Arithmetic operations on series can affect their convergence.
• Addition and subtraction can be performed on convergent series.
• What is the derivative of the power series $\sum_{n = 0}^{\infty} x^{n}$?


  $\sum_{n = 1}^{\infty} nx^{n - 1}$
• A divergent series has partial sums that grow indefinitely
• Match the convergence test with its applicability:

  Ratio test ↔️ Used for absolute convergence

  Integral test ↔️ Used for positive terms

  Alternating series test ↔️ Used for alternating series
• When combining series, the starting indices must be the same.
• What happens if one series converges and the other diverges when combined?

  The combined series diverges
• Arithmetic operations on series involve adding, subtracting, or multiplying series by constants
• Convergent series can be added term-by-term, and the resulting sum is the sum of individual sums.
• Match the arithmetic operation with its definition:

  Addition ↔️ $\sum a_{n} +$$\sum b_{n} =$$\sum (a_{n} +$$b_{n})$

  Subtraction ↔️ $\sum a_{n} - \sum b_{n} =$$\sum (a_{n} - b_{n})$

  Constant Multiplication ↔️ $k \sum a_{n} =$$\sum (ka_{n})$
• A series is said to be convergent if its partial sums approach a finite limit
• Addition and subtraction can only be performed on convergent series.
• Term-by-term differentiation and integration of power series do not change the radius of convergence
• Steps to combine two series using addition or subtraction:

  1️⃣ Check starting indices

  2️⃣ Adjust indices if necessary

  3️⃣ Combine series term-by-term
• If one series converges and the other diverges, the combined series diverges.
• Applying series manipulations involves arithmetic operations, index adjustments, and term-by-term operations
• When series start at different indices, they must be adjusted to align.
• Match the convergence test with its key feature:

  Ratio Test ↔️ Useful for factorials

  Root Test ↔️ Effective for nth powers

  Limit Comparison Test ↔️ Compares to known series
• What condition must $\lim_{n \to \infty} \frac{a_{n}}{b_{n}}$ satisfy in the Limit Comparison Test to conclude that $\sum a_{n}$ and $\sum b_{n}$ both converge or diverge?


  $0 < c < \infty$
• The Ratio Test is ideal for series with terms raised to the $n^{th}$ power.


  False
• Scalar multiplication of a convergent series by a constant scales its sum.
• Match the property with its description:

  Convergence ↔️ Partial sums approach a finite limit

  Divergence ↔️ Partial sums do not approach a finite limit

  Arithmetic Operations ↔️ Combine series using addition, subtraction, or constant multiplication

  Term-by-Term Operations ↔️ Apply differentiation or integration to power series
• Steps to combine two infinite series with different starting indices

  1️⃣ Ensure series have the same starting index

  2️⃣ Adjust indices if necessary

  3️⃣ Combine series