Let be the sequence of partial sums for the infinite series

MATHEMATICS · Multi-part · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series

Problem

Let {Sn}\{S_n\} be the sequence of partial sums for the infinite series ∑n=1∞an\sum_{n=1}^{\infty} a_n. The series is said to converge to SS provided that lim⁡n→∞Sn=S\displaystyle \lim_{n\to\infty} S_n = S. Throughout this question, assume the limit exists. Answer the following in complete sentences, using precise mathematical language. When you cite definitions or theorems, state them clearly before applying them. (a) State the formal ε\varepsilon–NN criterion that expresses lim⁡n→∞Sn=S\displaystyle \lim_{n\to\infty} S_n = S. (b) Explain why the existence of such an NN for every ε>0\varepsilon>0 guarantees that no other real number T≠ST\neq S can satisfy the same ε\varepsilon–NN condition with the sequence {Sn}\{S_n\}. (c) Interpret the inequality ∣S−SN∣<0.001|S-S_N|<0.001 in the context of approximating the sum of the series. What does this tell you about using SNS_N as an estimate for SS?

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