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- https://math.libretexts.org/Courses/City_College_of_San_Francisco/CCSF_Calculus_II__Integral_Calculus_._Lockman_Spring_2024/04%3A_Sequences_and_Series/4.01%3A_SequencesThis section introduces sequences, defining them as ordered lists of numbers generated by functions with natural numbers as inputs. It covers various types of sequences, including arithmetic and geome...This section introduces sequences, defining them as ordered lists of numbers generated by functions with natural numbers as inputs. It covers various types of sequences, including arithmetic and geometric, and explains how to represent sequences explicitly and recursively. The section also discusses limits of sequences and provides examples to illustrate how sequences behave, helping readers understand convergence and divergence.
- https://math.libretexts.org/Courses/Cosumnes_River_College/Math_401%3A_Calculus_II_-_Integral_Calculus/03%3A_Sequences_and_Series/3.02%3A_SequencesThis section introduces sequences, defining them as ordered lists of numbers generated by functions with natural numbers as inputs. It covers various types of sequences, including arithmetic and geome...This section introduces sequences, defining them as ordered lists of numbers generated by functions with natural numbers as inputs. It covers various types of sequences, including arithmetic and geometric, and explains how to represent sequences explicitly and recursively. The section also discusses limits of sequences and provides examples to illustrate how sequences behave, helping readers understand convergence and divergence.
- https://math.libretexts.org/Bookshelves/Algebra/Intermediate_Algebra_1e_(OpenStax)/12%3A_Sequences_Series_and_Binomial_Theorem/12.04%3A_Geometric_Sequences_and_Series\(\begin{aligned} S_{n}&= a_{1}+a_{1} r+a_{1} r^{2}+a_{1} r^{3}+\ldots+a_{1} r^{n-1} \\ r S_{n} &= a_{1} r+a_{1} r^{2}+a_{1} r^{3}+\ldots+a_{1} r^{n-1}+a_{1} r^{n}\\\hline S_{n}-r S_{n} &= a_{1} -a_{1...Sn=a1+a1r+a1r2+a1r3+…+a1rn−1rSn=a1r+a1r2+a1r3+…+a1rn−1+a1rnSn−rSn=a1−a1rn where a1 is the first term and r is the common ratio.Infinite Geometric Series: An infinite geometric series is an infinite sum whose first term is a1 and common ratio is r and is written
- https://math.libretexts.org/Courses/Cosumnes_River_College/Math_401%3A_Calculus_II_-_Integral_Calculus_Lecture_Notes_(Simpson)/03%3A_Sequences_and_Series/3.01%3A_SequencesIn this section, we introduce sequences and define what it means for a sequence to converge or diverge. We show how to find limits of sequences that converge, often by using the properties of limits f...In this section, we introduce sequences and define what it means for a sequence to converge or diverge. We show how to find limits of sequences that converge, often by using the properties of limits for functions discussed earlier. We close this section with the Monotone Convergence Theorem, a tool we can use to prove that certain types of sequences converge.
- https://math.libretexts.org/Courses/City_College_of_San_Francisco/CCSF_Calculus/08%3A_Sequences_and_Series/8.02%3A_SequencesThis section introduces sequences, defining them as ordered lists of numbers generated by functions with natural numbers as inputs. It covers various types of sequences, including arithmetic and geome...This section introduces sequences, defining them as ordered lists of numbers generated by functions with natural numbers as inputs. It covers various types of sequences, including arithmetic and geometric, and explains how to represent sequences explicitly and recursively. The section also discusses limits of sequences and provides examples to illustrate how sequences behave, helping readers understand convergence and divergence.
- https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/Gentle_Introduction_to_the_Art_of_Mathematics_(Fields)/03%3A_Proof_Techniques_I/3.02%3A_More_Direct_ProofsIn creating a direct proof, we need to look at our hypotheses, consider the desired conclusion, and develop a strategy for transforming A into B. Quite often you’ll find it easy to make several deduct...In creating a direct proof, we need to look at our hypotheses, consider the desired conclusion, and develop a strategy for transforming A into B. Quite often you’ll find it easy to make several deductions from the hypotheses, but none of them seems to be headed in the direction of the desired conclusion. The usual advice at this stage is “Try working backwards from the conclusion.”
- https://math.libretexts.org/Workbench/Intermediate_Algebra_2e_(OpenStax)/12%3A_Sequences_Series_and_Binomial_Theorem/12.04%3A_Geometric_Sequences_and_SeriesS n = a 1 + a 1 r + a 1 r 2 + a 1 r 3 + … + a 1 r n − 1 r S n = a 1 r + a 1 r 2 + a 1 r 3 + … + a 1 r n − 1 + a 1 r n ____________________________________________________ S n − r S n = a 1 −a 1 r n S ...S n = a 1 + a 1 r + a 1 r 2 + a 1 r 3 + … + a 1 r n − 1 r S n = a 1 r + a 1 r 2 + a 1 r 3 + … + a 1 r n − 1 + a 1 r n ____________________________________________________ S n − r S n = a 1 −a 1 r n S n = a 1 + a 1 r + a 1 r 2 + a 1 r 3 + … + a 1 r n − 1 r S n = a 1 r + a 1 r 2 + a 1 r 3 + … + a 1 r n − 1 + a 1 r n ____________________________________________________ S n − r S n = a 1 −a 1 r n
