``Sigma'' notation (C)

We need a ``shorthand'' to represent sums with more than a few terms. If f(n) is a rule which assigns the number f(n) to each integer n in the set of integers $\{K,K+1,\dots L\}$, such as f(n) = 2ⁿ, or f(n) = 2n+6, etc, then

$$\sum_{n=K}^L f(n)$$

denotes the sum of the values $f(K), f(K+1), \dots, f(L)$, and is read ``The sum of f(n) from n = K to L''. For example

$$\sum_{n=2}^5 (2n+1) = (2\cdot 2 + 1) + (2\cdot 3 +1)+ (2\cdot 4 +1)+ (2\cdot 5 +1) = 5 + 7 + 9 + 11 = 32$$

and

$$\sum_{k=-2}^5 2^k = 2^{-2} + 2^{-1} + 2^0 + 2^1 +2^2+2^3+2^4+2^5 = \frac{3}{4}+63.$$

Exercises Evaluate each of the following:

1.

$${\sum_{j=-2}^4 (2j-3)}$$;

2.

$${\sum_{j=0}^5 j^2}$$;

3.

$${\sum_{j=-8}^{-2} \sin(j\pi/6)}$$;

4.

$${\sum_{a=0}^{4} (1/2)^a}$$;