Learn how to find the sum of arithmetic and geometric series when they're presented in sigma (summation) notation! Mario's Math Tutoring guides you through six challenging examples, covering both finite and infinite series. Discover how to identify if a series is arithmetic or geometric, correctly determine the number of terms (n), find the first term (a₁), and apply the appropriate sum formulas. We'll also explore special cases like infinite geometric series and divergent series. In this video, you'll discover: 0:00 Introduction to Sum of Arithmetic & Geometric Series (Sigma Notation) 0:20 Understanding Sigma (Summation) Notation 0:50 Example 1: Arithmetic Series Σ (2i + 5) from i=1 to 8 1:00 Writing out the first few terms 1:40 Identifying as an Arithmetic Series 2:00 Arithmetic Series Sum Formula: Sn = n/2 * (a₁ + an) 2:20 Finding the number of terms (n), first term (a₁), and last term (an) 2:50 Calculating the Sum (112) 3:20 Example 2: Geometric Series Σ (10 * (-1/2)^(i-1)) from i=1 to 10 3:40 Writing out the first few terms 4:10 Identifying as a Geometric Series and finding the common ratio (r) 4:40 Finite Geometric Series Sum Formula: Sn = a₁ * (1 - rⁿ) / (1 - r) 5:30 Calculating the Sum (1705 / 256 or ~ ) 6:50 Example 3: Arithmetic Series Σ (4 - 3i) from i=5 to 13 7:00 Finding the first term (using i=5) and last term (using i=13) 7:40 Identifying as an Arithmetic Series 8:00 Correctly calculating the number of terms (n = last - first + 1) 8:50 Calculating the Sum (-207) 9:20 Example 4: Infinite Geometric Series Σ (2 * (2/3)^(i-1)) from i=3 to ∞ 9:40 Writing out the first few terms 10:10 Identifying as a Geometric Series and finding the common ratio (r) 10:30 Infinite Geometric Series Sum Formula: S∞ = a₁ / (1 - r) (Conditions for convergence) 11:00 Calculating the Sum (8/3) 11:50 Example 5: Finite Geometric Series Σ (-2 * 3^(i-1)) from i=1 to 9 12:10 Writing out the first few terms 12:40 Identifying as a Geometric Series and finding the common ratio (r) 13:00 Finding the number of terms (n) 13:20 Applying the Finite Geometric Series Sum Formula 14:00 Calculating the Sum (19,682) 14:20 Example 6: Infinite Geometric Series Σ (3/2)^(i-1) from i=0 to ∞ 14:40 Identifying as an Infinite Geometric Series and finding the common ratio (r) 15:00 Checking for Convergence (Ratio not between -1 and 1) 15:20 Conclusion: No Sum (Divergent Series) Master summation notation and confidently calculate the sums of various series! #SigmaNotation #Summation #ArithmeticSeries #GeometricSeries #InfiniteSeries #Algebra2 #Precalculus #MathTutorial #MathHelp ➡️JOIN the channel as a CHANNEL MEMBER at the "ADDITIONAL VIDEOS" level to get access to my math video courses(Algebra 1, Algebra 2/College Algebra, Geometry, and PreCalculus), midterm & final exam reviews, ACT and SAT prep videos and more! (Over 390+ videos)