Chapter B, Problem 32AYU

In Problems 29-32, find the first five terms in the sequence of partial sums for each series.

${\displaystyle \underset{k=1}{\overset{\infty }{\sum }}{k}^2}$

To determine

To find: The first five terms in the sequence of partial sums for each series.

Answer to Problem 32AYU

Solution:

The first five terms in the sequence of partial sums for each series is $1, 5, 14, 30, 55$.

Explanation of Solution

Given:

${\displaystyle {\sum }_{k = 1}^{\infty }{k}^2}$

Calculation:

${\displaystyle \sum _{k = 1}^{\infty }{k}^2 = 1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2 + 7^2 + \cdots \mathrm{....}}$

${\displaystyle \sum _{k = 1}^{\infty }{k}^2 = 1 + 4 + 9 + 16 + 25 + 36 + 49 + \cdots }$

The first five terms in the sequence of partial sums for each series is

$\begin{array}{l}1\\ 1 + 4 = 5\\ 1 + 4 + 9 = 14\\ 1 + 4 + 9 + 16 = 30\\ 1 + 4 + 9 + 16 + 25 = 55\end{array}$

The first five terms in the sequence of partial sums for each series is $1, 5, 14, 30, 55$.