Answer:
No solution
Step-by-step explanation:
We have
[tex]$\sum_{x=8}^{4}x^2 + 9\left(\dfrac{\cos^2(\infty)}{\sin^2(\infty)} \right)$[/tex]
For the sum it is not correct to assume
[tex]$\sum_{x=8}^{4}x^2= 8^2 + 7^2+6^2+5^2+4^2 = 64+49+36+25+16 = 190$[/tex]
Note that for
[tex]$\sum_{x=a}^b f(x)$[/tex]
it is assumed [tex]a\leq x \leq b[/tex] and in your case [tex]\nexists x\in\mathbb{Z}: a\leq x\leq b[/tex] for [tex]a>b[/tex]
In fact, considering a set [tex]S[/tex] we have
[tex]$\sum_{x=a}^b (S \cup \varnothing) = \sum_{x=a}^b S + \sum_{x=a}^b \varnothing $[/tex] that satisfy [tex]S = S \cup \varnothing[/tex]
This means that, by definition [tex]\sum_{x=a}^b \varnothing = 0[/tex]
Therefore,
[tex]$\sum_{x=8}^{4}x^2 = 0$[/tex]
because the sum is empty.
For
[tex]9\left(\dfrac{\cos^2(\infty)}{\sin^2(\infty)} \right)[/tex]
we have other problems. Actually, this case is really bad.
Note that [tex]\cos^2(\infty)[/tex] has no value. In fact, if we consider for the case
[tex]$\lim_{x \to \infty} \cos^2(x)$[/tex], the cosine function oscillates between [tex][-1, 1][/tex] , and therefore it is undefined. Thus, we cannot evaluate
[tex]9\left(\dfrac{\cos^2(\infty)}{\sin^2(\infty)} \right)[/tex]
and then
[tex]$\sum_{x=8}^{4}x^2 + 9\left(\dfrac{\cos^2(\infty)}{\sin^2(\infty)} \right)$[/tex]
has no solution
