#### To determine

The rank of the three quantities are
∫16f(x)dx,∑i=15ai
and
∑i=26ai
.

#### Answer

The increasing order of the rank of the three quantities is,
∑i=26ai<∫16f(x)dx<∑i=15ai
.

#### Explanation

**Result used**: Remainder Estimate for the Integral Test

If the function
f(x)
is continuous, positive, and decreasing for
x≥n
and
∑an
is convergent and
Rn=s−sn
, then
∫n+1∞f(x)dx≤Rn≤∫n∞f(x)dx
.

**Given**:

The function
f(x)
is continuous, positive, and decreasing for
n≥1
and
an=f(n)
.

**Calculation**:

From the graph below, the total area of the rectangles is more than the area under
f(x)
of the curve.

From Figure 1 and the result stated above, it can be concluded that
∫16f(x)dx<∑i=15ai
.

From the graph below, the area of the rectangles is less than the area under
f(x)
of the curve.

From Figure 2, it can be concluded that
∑i=26ai<∫16f(x)dx
.

Combine the inequalities obtained from Figure (1) and Figure (2) and get the final rank order as,
∑i=26ai<∫16f(x)dx<∑i=15ai
.