### Home > CALC > Chapter 2 > Lesson 2.1.1 > Problem2-11

2-11.

Expand and evaluate each of the following sums.

1. $\displaystyle \sum _ { n =-4 } ^ { 4 }n^2$

$(-4)^² +$ $(-3)^² +$ $(-2)^² +$ $(-1)^² + (0)^²$ $+ (1)^² +$ $(2)^² + (3)^²$ $+ (4)^² =$ ___

or, because of the symmetry of $n^2$, this problem can be simplified as:$=2\sum_{-4}^{0}n^{2}=2\left ( (-4)^{2}+(-3)^{2} +(-2)^{2}+(-1)^{2}\right )=2(30)=60$

1. $\displaystyle \sum _ { n =-4 } ^ { 4 }k^3$

Think about the symmetry of odd functions such as $k^³$.
Justify why the sum from $−4$ to $+4$ will be $0$.

1. $\displaystyle \sum _ { j =-3 } ^ { 3 }2^j$

Unfortunately, $2^j$ does not have symmetry across the $y$-axis. We have to expand and evaluate all terms.
$= 2^{ −3} + 2^{ −2} + ... + 2^2 + 2^1 =$ _______

1. $\displaystyle \sum _ { i =-5 } ^ { 5 }\text{sin }i$

sin $x$ is an odd function. Consider symmetry before evaluating.