Expand and evaluate each of the following sums.

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

   Hint (a):

   or, because of the symmetry of $n^2$, this problem can be simplified as:

   $\displaystyle =2\sum_{-4}^{0}n^{2}=2\left ( (-4)^{2}+(-3)^{2} +(-2)^{2}+(-1)^{2}\right )=2(30)=60$

   Steps (a):

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

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

   Hint (b):

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

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

   Hint (c):

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

4. $\displaystyle\sum _ { i = - 5 } ^ { 5 } \operatorname { sin } ( i )$

   Hint (d):

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