A series is the addition of the terms of an ordered sequence, often written $\sum_{k=0}^{\infty}a_k$.

What is the $n$th partial sum?

It is the finite running total $s_n=\sum_{k=0}^{n}a_k=a_0+a_1+\cdots+a_n$.

What does convergence mean for a series?

The series converges when its partial sums approach a finite limit. That limit is the sum of the series.

How are terms related to partial sums?

For $n\geq1$, the term is the difference between consecutive partial sums: $a_n=s_n-s_{n-1}$.

What is truncation error?

It is the remaining tail after a finite partial sum: $s-s_n=\sum_{k=n+1}^{\infty}a_k$.

Real analysisLesson 1 of 2 in the course Riemann integration and series

Infinite Series: Convergence, Operations, Tests, and Function Series

5 flashcards · 5 quiz questions · 5 match pairs · 5 fill-in sentences · English

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Summary

An infinite series describes the addition of the terms of an ordered sequence. Since infinitely many additions cannot be completed one at a time, calculus defines a series through the limit of its finite partial sums.

$a_0+a_1+a_2+\cdots=\sum_{k=0}^{\infty}a_k.$

The index may begin at zero or one, depending on how the sequence is defined. Terms need not be ordinary numbers: in suitable settings, they can be functions, matrices, or other objects that can be added.

All flashcards

What is a series?: A series is the addition of the terms of an ordered sequence, often written $\sum_{k=0}^{\infty}a_k$ .
What is the $n$ th partial sum?: It is the finite running total $s_n=\sum_{k=0}^{n}a_k=a_0+a_1+\cdots+a_n$ .
What does convergence mean for a series?: The series converges when its partial sums approach a finite limit. That limit is the sum of the series.
How are terms related to partial sums?: For $n\geq1$ , the term is the difference between consecutive partial sums: $a_n=s_n-s_{n-1}$ .
What is truncation error?: It is the remaining tail after a finite partial sum: $s-s_n=\sum_{k=n+1}^{\infty}a_k$ .

Quiz questions

What is a partial sum of a series?Answer: A finite sum of an initial group of terms.
When does an infinite series converge?Answer: When its partial sums approach a finite limit.
What is the sum of a convergent series?Answer: The limit of its sequence of partial sums.
How can $a_n$ be recovered from partial sums for $n\geq1$ ?Answer: Subtract consecutive partial sums: $a_n=s_n-s_{n-1}$ .
What is the truncation error when $s_n$ approximates the sum $s$ ?Answer: The remaining tail, $s-s_n=\sum_{k=n+1}^{\infty}a_k$ .

Based on the Wikipedia article Series (mathematics), licensed CC BY-SA 4.0. Adapted into a summary, flashcards and quizzes.

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