Let be a continuous

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

Problem

Let

f

be a continuous, positive, and decreasing function on

[1,\infty)

. Define

a_k=f(k)

for each integer

k\ge 1

, and let

S_n=\sum_{k=1}^{n} a_k \quad \text{and} \quad R_n=\sum_{k=n+1}^{\infty} a_k.

The series

\sum_{k=1}^{\infty} a_k

is known to converge. (a) Describe, in words, the geometric meaning of

S_n

and

R_n

when the graph of

y=f(x)

is interpreted as a collection of rectangles of width

1

on successive integer intervals. (b) Using the geometric interpretation from part (a), explain why

R_n

is less than the area under the curve

y=f(x)

from

x=n

x=\infty

, and why it is greater than the area under the curve from

x=n+1

x=\infty

. (c) Explain why the difference between these two bounding areas must decrease (that is, the bounds become tighter) as

n

increases.