10.15 Manipulating Series

• What is a sequence in mathematical terms?

  An ordered list of numbers
• What is an arithmetic series?

  Constant difference between terms
• What is a geometric series?

  Constant ratio between terms
• A series diverges if its sum approaches a finite value as more terms are added.

  False
• An alternating series is a series where the terms alternate between positive and negative.
• A telescoping series is one where each term cancels out with the previous
• An alternating series converges if the absolute value of the terms decreases
• The Integral Test states that if the improper integral $\int_{1}^{\infty} f(x) dx$ diverges, then the series $\sum_{n = 1}^{\infty} f(n)$ also diverges.


  True
• What is the notation for a series?

  $\sum_{n = 1}^{\infty} a_{n}$
• What is the formula for the sum of a geometric series with n terms, first term a₁, and common ratio r?

  S_{n} = a_{1} \cdot \frac{1 - r^{n}}{1 - r}</latex>
• What does the convergence or divergence of a series refer to?

  Whether the sum approaches a finite value
• What condition must positive terms satisfy for a series to converge?

  $\lim_{n \to \infty} a_{n} =$$0$
• A mixed sign series converges if both its positive and negative parts converge
• A series with positive terms diverges if the limit of its terms is not zero.

  True
• A series with mixed signs converges if both its positive and negative parts converge
• A series with positive terms converges if the limit of its terms is zero
• An alternating series converges if the absolute values of its terms decrease to 0.


  True
• The sum of a telescoping series is given by the formula a_{1} - \lim_{n \to \infty} a_{n}
• The sum of a telescoping series is always finite.

  True
• The example alternating series $\sum_{n = 1}^{\infty} \frac{( - 1)^{n + 1}}{n}$ converges to \ln{2}
• A series is the sum of the terms in a sequence
• The sum of an arithmetic series is given by the formula $S_{n} =$$\frac{n}{2}(a_{1} +$$a_{n})$


  True
• The sum of a geometric series is given by the formula $S_{n} =$$a_{1} \cdot \frac{1 - r^{n}}{1 - r}$, where r is the common ratio.
• Under what condition does a series with positive terms converge?

  $\lim_{n \to \infty} a_{n} =$$0$
• What test is used to determine convergence or divergence by comparing the series to a known converging or diverging series?

  Comparison test
• How is the sum of a telescoping series calculated?

  Difference between first and last terms
• What are the two conditions for the convergence of an alternating series according to the alternating series test?

  Absolute value decreases and limit is 0
• A sequence is an ordered list of numbers, each called a term
• The sum of an arithmetic series with n terms is given by $S_{n} =$$\frac{n}{2}(a_{1} +$$a_{n})$.


  True
• In an arithmetic series, the difference between consecutive terms is constant
• A series diverges if its sum grows without bound as more terms are added.


  True
• What does it mean for a series to converge?

  Sum approaches a finite value
• What is the purpose of the comparison test in determining series convergence?

  Compares to known series
• What type of series has successive terms that cancel each other out?

  Telescoping series
• What is the key difference between telescoping and alternating series in terms of term relationship?

  Terms cancel vs. alternate signs
• Match the series type with its convergence property:

  Telescoping Series ↔️ Converges if $a_{n}$ approaches a finite limit

  Alternating Series ↔️ Converges if absolute values decrease to 0
• What is the test used to determine the convergence of an alternating series?

  Alternating series test
• A telescoping series converges if its terms $a_{n}$ approach a finite limit.


  True