2 - Calculus

Question

Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading. 

(a) A sum that would not be suitable for expressing in sigma notation. 

(b) Two different sigma notation expressions of the same sum. 

(c) A sum from k = 1 to k = n that is smaller for n = 10 than it is for n = 5 

Step-by-Step Solution

Answer

Part(a): The sum of numbers $1+2+8+9+15+16+.....$

Part(b): The sum of numbers $1+2+3+4+\dots .+1^2+2^2+3^2+4^2+\dots ..$

Part(c): $\sum _{k=1}^n-k$

Part (a): Explanation for the statement.

A sum that would be difficult to describe in sigma notation.

An example for the sum of numbers $1+2+8+9+15+16+\cdots$ cannot be expressed using sigma notation.

Since there is a break between the sum of the series in certain places sigma notation is not useful to express this example.

Part (b): Explanation for the statement.

Two different sigma notation expressions of the same sum.

The example for the sum of numbers $1+2+3+4+\dots .+1^2+2^2+3^2+4^2+\dots ..$can be expressed using two sigma notations as shown below$\sum _{k=1}^{\infty }\left(k+k^2\right)$.

Part (c): Explanation for the statement.

A sum from $k=1$ to $k=n$that is smaller for $n=10$ than it is for $n=5$ 

Note the sigma sum of numbers given below.

$\sum _{k=1}^n-k$

And, 

$\sum _{k=1}^5-k=-1+-2+-3+-4+-5=-15$

And,

$\sum _{k=1}^{10}-k=-1+-2+-3+-4+-5+-6+-7+-8+-9+-10=-55$

Thus, the sum$\sum _{k=1}^n-k$ is smaller for $n=10$ than it is for $n=5$.