AP Calculus BC Flashcards: Defining Convergent and Divergent Infinite Series
Written by AP Content Team, Verified for 2026 AP Exams, Last updated: May 2026
Review key ideas with interactive flashcards. This set includes 10 cards to help you master important concepts.
The sequence of partial sums for a series is $S_n$. If $\lim_{n \to \infty} S_n = 10$, what can you conclude about the series?
The series converges, and its sum is 10.
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The sequence of partial sums for a series is $S_n$. If $\lim_{n \to \infty} S_n = 10$, what can you conclude about the series?
The series converges, and its sum is 10.
What is the sole condition under which an infinite series is said to have a sum S?
The series has a sum S if and only if the limit of its sequence of partial sums exists and is equal to S.
How is the convergence of an infinite series determined by its partial sums?
A series converges if its sequence of partial sums approaches a finite limit; otherwise, the series diverges.
If the limit of the sequence of partial sums for a series does not exist, what is true about the series?
The series diverges because the limit of its partial sums must exist for it to converge.
What is the 'sum' of a convergent infinite series?
The sum of a convergent infinite series is the real number S that is the limit of its sequence of partial sums.
A series has a sequence of partial sums given by $S_n = 2 - \frac{1}{n}$. Does this series converge or diverge, and if it converges, what is its sum?
The series converges because the limit of its partial sums as n approaches infinity is 2. The sum of the series is 2.
What is the nth partial sum of a series?
The nth partial sum is the sum of the first n terms of a series.
Explain the relationship between an infinite series and its corresponding sequence of partial sums.
The sequence of partial sums represents the cumulative sum of the series' terms up to n. The behavior of this sequence as n approaches infinity determines whether the infinite series itself converges or diverges.
What is the primary objective when analyzing an infinite series?
The primary objective is to determine whether the series converges to a finite sum or diverges.
Define what it means for an infinite series to converge.
An infinite series converges to a sum S if and only if the limit of its sequence of partial sums exists and is equal to S.