Question

Explain why terms in the sum in Example 6 with n equals to 4 are completely different from the terms in the sum when n equals to 3. How can the sum from $k=1\mathrm{to}k=4$ be smaller than the sum from $k=1tok=3$? What will happen as n gets larger in this example?

Short answer

The variable n is in the denominator. Hence, the sum with n equals to 4 are completely different from the terms in the sum when n equals to 3.

If n gets larger, the denominator will be much larger and hence sum will be almost 0.

Step-by-step solution

Step 1. Given Information

The given expression is$\sum _{k=1}^n\frac{k^2}{n^3}$

Step 2. Explanation

Since, the variable n is in the denominator of the given sum expression, hence the sum with n equals to 4 are completely different from the terms in the sum with n equals to 3.

The sum $k=1tok=4is\frac{15}{32}\approx 0.46875$and the sum from $k=1tok=3is\frac{14}{27}\approx 0.5185$

Each of the terms in second sum has larger denominator than that of the first sum.

So, each of the terms in second sum is much smaller than that of each of the terms in the first sum.

Thus, the sum from $k=1tok=4$be smaller than the sum from $k=1tok=3$

If n gets larger the denominator will be much larger than the numerator and hence the sum will be almost zero.