Partial sums of arithmetic series

Learning Objectives Define an arithmetic series and its partial sum, S_n. Apply the formula for the nth partial sum of an arithmetic series. Express the partial sum formula, S_n, as a quadratic function of n. Set up and calculate the limit of a partial sum of an arithmetic series as n approaches infinity. Explain why the limit of a partial sum of any non-trivial arithmetic series is infinite. Determine if the limit is positive or negative infinity based on the common difference. If you saved $5 today, $8 tomorrow, $11 the next day, and so on, how much money would you have after a year? What about... forever? 📈 This tutorial connects the familiar concept of arithmetic series with the powerful tool of limits. You will learn how to find the formula for the sum of the first &#...

TermDefinitionExample Arithmetic SequenceA sequence of numbers where the difference between consecutive terms is constant.The sequence 5, 9, 13, 17, ... is an arithmetic sequence because each term is 4 more than the previous one. Common Difference (d)The constant value added to each term to get the next term in an arithmetic sequence.In the sequence 5, 9, 13, 17, ..., the common difference, d, is 4. Arithmetic SeriesThe sum of the terms in an arithmetic sequence.For the sequence 5, 9, 13, 17, ..., the corresponding arithmetic series is 5 + 9 + 13 + 17 + ... Partial Sum (S_n)The sum of the first 'n' terms of a series. It is a sequence of sums.For the series 5 + 9 + 13 + ..., the 3rd partial sum is S_3 = 5 + 9 + 13 = 27. Limit at InfinityThe value that a function or sequence appro...

Formula for the nth Partial Sum S_n = \frac{n}{2}(2a_1 + (n-1)d) This is the primary formula used to calculate the sum of the first 'n' terms of an arithmetic series, where 'a_1' is the first term, 'n' is the number of terms, and 'd' is the common difference. Quadratic Form of the Partial Sum S_n = (\frac{d}{2})n^2 + (a_1 - \frac{d}{2})n By algebraically expanding the primary formula, we can see that S_n is a quadratic function of n. This form is extremely useful for evaluating limits, as the behavior at infinity is determined by the highest-power term, n^2. Limit of the Partial Sum \lim_{n \to \infty} S_n = \lim_{n \to \infty} [(\frac{d}{2})n^2 + (a_1 - \frac{d}{2})n] To find the limit of the partial sum, we take the limit of...