Sigma notation 1

Instructions

Click the button below to start.
Parts (i)-(iii) will stay the same. Part (iv) will be changed each time you click the button again.
Practice until you can perform the "substitution" technique reliably.

Question

A sequence $u_n$ is defined by $\displaystyle u_n = \frac{1}{n^3}$ for $n \in \mathbb{Z}, n \geq 1$ .

Show that $\displaystyle u_n - u_{n+1} = \frac{3n^2+3n+1}{n^3(n+1)^3}$ .
Hence find $\displaystyle \sum_{n=1}^N \frac{3n^2+3n+1}{n^3(n+1)^3}$ .
Give a reason why the series in part (ii) is convergenet and state the sum to infinity.
Use your answer to part (ii) to find
$1$

Answer

$\displaystyle 1 - \frac{1}{(N+1)^3}$ .
As $N \to \infty$ , $\displaystyle \frac{1}{(N+1)^3} \to 0$ . $1 - \displaystyle \frac{1}{(N+1)^3} \to 1$ so the series is convergent and the sum to infinity is $1$ .
$2$

a dot b

A Mathematics Website

Sigma notation 1

Instructions

Question

Answer