MATHEMATICS · Multi-part · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series
Problem
Consider the infinite series S=n=1∑∞n2+3n1. A student computes successive partial sums and records
S4=0.6300 and S5=0.6715 (rounded to four decimal places).
Answer the following questions related to numerical approximation of S.
(a) Describe, in words, what the value of a partial sum SN represents in relation to the exact value S of the series.
(b) Given the values of S4 and S5 above, explain what inequalities you can state about S. Justify your answer without calculating additional terms.
(c) Write a clear plan (no numerical execution required) for deciding how many terms of the series must be added so that the approximation differs from S by less than 0.0005.