How to estimate the value of a series ?

 Let's estimate the value of a convergent series :

We can write this series as:

From the equation above, the reminder can be written as:

Let's use the integral test to estimate the remainder. We are going to compare the reminder to the improper integral of a function f. The remainder can be written as a sum of areas of rectangles as follows:

The length of each rectangle is considered to be 1.

We consider two scenarios. The first scenario is considered in the following figure:

In this case the sum of the areas of the rectangles that represents the reminder Rₙ is inferior to the improper integral whose limits are N and +∞. We write:

Let's consider the second scenario in the following figure:

 

The reminder Rₙ is superior to the improper integral and we can write:

We can combine the two comparisons of  Rₙ by writing:

The preceding inequality represents an approximation of the remainder.

Example

Solution

a) S₁₀  = 1/1³ + 1/2³ + 1/3³ + 1/4⁴ + .....                 + 1/10³≃ 1.19753

Estimate the error means estimate the remainder. According to the formula for the estimation of the remainder:

b) Estimating the series within 0.001 means the error or the remainder should be less than 0.001.

Then 1/2N²  < 0.001

2N² > 1000⇒ N² > 500 ⇒ N>10⎷5 ⇒N>22.36

By rounding N to the nearest hundreth, we can choose the least value of N as N = 23.  

Practice