Partial Sum

Definition

The partial sum of a sequence is the sum of the first $n$ terms of the sequence. It represents the accumulation of the sequence up to a certain point, providing a way to understand the overall behavior and convergence of the sequence.

5 Must Know Facts For Your Next Test

1. The partial sum of a sequence $a_1, a_2, a_3, \dots, a_n$ is the sum of the first $n$ terms, denoted as $S_n = a_1 + a_2 + a_3 + \dots + a_n$.
2. Partial sums are used to analyze the behavior of a sequence and determine whether the sequence converges or diverges.
3. In the context of geometric sequences, the partial sum formula is $S_n = a(1 - r^n) / (1 - r)$, where $a$ is the first term and $r$ is the common ratio.
4. The behavior of the partial sums of a geometric sequence can be used to determine the convergence or divergence of the corresponding geometric series.
5. Partial sums play a crucial role in understanding the properties of infinite series, such as the convergence or divergence of the series.

Review Questions

• Explain the relationship between partial sums and the convergence of a sequence.
  • The behavior of the partial sums of a sequence is directly related to the convergence or divergence of the sequence. If the partial sums of a sequence approach a finite limit as the number of terms increases, then the sequence is said to converge. Conversely, if the partial sums of a sequence continue to increase or decrease without bound, then the sequence is said to diverge. Analyzing the pattern and behavior of the partial sums is a key way to determine the convergence or divergence of a sequence.
• Describe the formula for the partial sum of a geometric sequence and explain how it can be used to analyze the convergence of the corresponding geometric series.
  • The formula for the partial sum of a geometric sequence with first term $a$ and common ratio $r$ is $S_n = a(1 - r^n) / (1 - r)$. This formula allows us to easily calculate the partial sums of the sequence and observe their behavior as the number of terms $n$ increases. If the common ratio $|r| < 1$, then the partial sums will converge to a finite limit, and the corresponding geometric series will also converge. If $|r| \geq 1$, then the partial sums will diverge, and the geometric series will also diverge.
• Explain how partial sums are used to understand the properties of infinite series, and discuss the significance of this concept in the study of sequences and series.
  • Partial sums are fundamental to the study of infinite series, as they provide a way to analyze the behavior and convergence of the series. By examining the pattern and limit of the partial sums, we can determine whether an infinite series converges to a finite value or diverges. This understanding of partial sums is crucial in the study of sequences and series, as it allows us to classify different types of series, apply convergence tests, and make predictions about the behavior of sequences and their corresponding series. The concept of partial sums is a unifying idea that connects the properties of sequences and the properties of series, making it a central and indispensable tool in the field of mathematical analysis.

Related terms

Sequence:

A sequence is an ordered list of numbers or mathematical objects, where each element is called a term of the sequence.

Series:

A series is the sum of the terms of a sequence, representing the accumulation of the sequence as the number of terms increases.

Convergence:

Convergence refers to the behavior of a sequence or series as the number of terms increases, specifically whether the sequence or series approaches a finite limit.