Introduction to partial sums

Learning Objectives Define an infinite series and its corresponding sequence of partial sums. Calculate the first 'n' partial sums for a given infinite series. Derive a general formula for the nth partial sum, S_n, for telescoping and geometric series. Define convergence of a series as the existence of a finite limit of its sequence of partial sums. Determine if a series converges or diverges by evaluating the limit of its nth partial sum. Calculate the sum of a convergent series by finding lim_{n→∞} S_n. Can you add an infinite number of positive values together and get a finite number? 🤔 It sounds impossible, but we're about to see how it's a cornerstone of calculus. This lesson introduces the concept of an infinite series, which is an infinite sum of...

TermDefinitionExample Infinite SeriesThe sum of the terms of an infinite sequence. It is denoted by \sum_{n=1}^{\infty} a_n.The series 1 + 1/2 + 1/4 + 1/8 + ... can be written as \sum_{n=1}^{\infty} (1/2)^{n-1}. Term of a Series (a_n)The nth number being added in an infinite series. It is determined by a formula dependent on 'n'.In the series \sum_{n=1}^{\infty} \frac{1}{n^2}, the third term (a_3) is 1/3^2 = 1/9. Partial Sum (S_n)The sum of the first 'n' terms of an infinite series. It is a finite sum that approximates the total infinite sum.For the series \sum_{k=1}^{\infty} k, the 4th partial sum is S_4 = 1 + 2 + 3 + 4 = 10. Sequence of Partial SumsThe sequence formed by all the partial sums of a series: S_1, S_2, S_3, ..., S_n, ...For the series \sum_{n=1}^{\infty}...

The nth Partial Sum Formula S_n = \sum_{i=1}^{n} a_i = a_1 + a_2 + a_3 + ... + a_n This is the fundamental definition of a partial sum. To find S_n, you add up all the terms of the series from the first term (a_1) up to the nth term (a_n). The Sum of an Infinite Series S = \lim_{n \to \infty} S_n The sum (S) of an infinite series is defined as the limit of its sequence of partial sums. If this limit exists and is a finite number, the series converges to S. If the limit does not exist or is infinite, the series diverges.