An infinite series is associated with its sequence of partial sums

MATHEMATICS · Subjective · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series

Problem

An infinite series ∑n=1∞an\displaystyle \sum_{n=1}^{\infty} a_n is associated with its sequence of partial sums Sn=∑k=1nakS_n=\sum_{k=1}^{n} a_k. (a) Define what it means for ∑n=1∞an\displaystyle \sum_{n=1}^{\infty} a_n to converge in terms of the sequence {Sn}\{S_n\}. (b) Assume that an>0a_n>0 for all nn and that {Sn}\{S_n\} is increasing and bounded above. Describe, with clear reasoning, what must happen to {Sn}\{S_n\} as n→∞n\to\infty and what that implies about the series. (c) Explain why an unbounded sequence of partial sums cannot correspond to a convergent infinite series, even if the individual terms ana_n approach 00.

Related theory

Browse the problem library