Q. 5 - Calculus

Question

Calculating sums: Determine the value of each of the sums that

follow. Some can be computed directly, some require the use

of sum formulas, and for some you will have to also compute

a limit .

$\underset{n\to \infty }{\lim }\sum _{k=1}^n\frac{k^2+1}{n^3}$

Step-by-Step Solution

Answer

The given sigma function is diverges .

Step 1. Given information .

Consider the given sigma function $\underset{n\to \infty }{\lim }\sum _{k=1}^n\frac{k^2+1}{n^3}$ .

Step 2. Apply the constant multiplication rule .

The constant multiplication rule is $\sum _{k=1}^{n}c·a_n=c·\sum _{k=1}^n{a}_n$.

Step 3. Find the value .

Applying the constant multiplication rule .

$$\begin{gathered} \Rightarrow \frac{1}{n^3}\underset{n\to \infty }{\lim }\sum _{k=1}^n\left(k^2+1\right) \\ \Rightarrow \frac{1}{n^3} diverges \end{gathered}$$