Partial sums of geometric series

Learning Objectives Define a geometric series and its partial sum. Derive and apply the formula for the nth partial sum of a geometric series. Identify the first term (a) and the common ratio (r) from a series presented in expanded or sigma notation. Determine whether an infinite geometric series converges or diverges based on its common ratio, r. Calculate the limit of the sequence of partial sums for a convergent geometric series. Distinguish between a finite partial sum (S_n) and the sum of an infinite series (S). Imagine a bouncing ball that always rebounds to 2/3 of its previous height. Will it ever stop bouncing, and what total distance does it travel? 🏀 Let's find out using the power of limits! In this tutorial, we will explore geometric series and their partia...

TermDefinitionExample Geometric SeriesThe sum of the terms of a geometric sequence. Each term is found by multiplying the previous term by a constant value called the common ratio (r).The sequence 4, 2, 1, 0.5, ... gives the series 4 + 2 + 1 + 0.5 + ... Partial Sum (S_n)The sum of the first 'n' terms of a series. It represents a finite portion of the entire series.For the series 4 + 2 + 1 + 0.5 + ..., the third partial sum is S_3 = 4 + 2 + 1 = 7. Common Ratio (r)The constant factor between consecutive terms in a geometric series. It is found by dividing any term by its preceding term (r = a_k / a_{k-1}).In the series 81 + 27 + 9 + ..., the common ratio is r = 27 / 81 = 1/3. ConvergenceAn infinite series is convergent if its sequence of partial sums (S_n) approaches a single, fin...

Formula for the nth Partial Sum S_n = a(1 - r^n) / (1 - r) Use this formula to find the sum of the first 'n' terms of a geometric series, where 'a' is the first term, 'r' is the common ratio (r ≠ 1), and 'n' is the number of terms. Condition for Convergence A geometric series converges if and only if |r| < 1. This is the essential test to determine if an infinite geometric series has a finite sum. If the absolute value of the common ratio is less than 1, the series converges. If |r| ≥ 1, it diverges. Limit of Partial Sums (Sum to Infinity) S = lim_{n->∞} S_n = a / (1 - r) If a series converges (|r| < 1), this formula gives the exact value it sums to. This is the limit of the partial sum formula, as the r^n term approac...