Mister Exam
Graphing y = ((-1)^n+1)/n
The solution
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n
(-1) + 1
f(n) = ---------
n
$$f{\left(n \right)} = \frac{\left(-1\right)^{n} + 1}{n}$$
f = ((-1)^n + 1)/n
The graph of the function
The domain of the function
The points at which the function is not precisely defined:
$$n_{1} = 0$$
$$n_{1} = 0$$
The points of intersection with the X-axis coordinate
Graph of the function intersects the axis N at f = 0
so we need to solve the equation:
$$\frac{\left(-1\right)^{n} + 1}{n} = 0$$
Solve this equation
The points of intersection with the axis N:
Analytical solution
$$n_{1} = 1$$
Numerical solution
$$n_{1} = -75$$
$$n_{2} = 27$$
$$n_{3} = 21$$
$$n_{4} = -79$$
so we need to solve the equation:
$$\frac{\left(-1\right)^{n} + 1}{n} = 0$$
Solve this equation
The points of intersection with the axis N:
Analytical solution
$$n_{1} = 1$$
Numerical solution
$$n_{1} = -75$$
$$n_{2} = 27$$
$$n_{3} = 21$$
$$n_{4} = -79$$
Extrema of the function
In order to find the extrema, we need to solve the equation
$$\frac{d}{d n} f{\left(n \right)} = 0$$
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
$$\frac{d}{d n} f{\left(n \right)} = $$
the first derivative
$$\frac{\left(-1\right)^{n} i \pi}{n} - \frac{\left(-1\right)^{n} + 1}{n^{2}} = 0$$
Solve this equation
Solutions are not found,
function may have no extrema
$$\frac{d}{d n} f{\left(n \right)} = 0$$
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
$$\frac{d}{d n} f{\left(n \right)} = $$
the first derivative
$$\frac{\left(-1\right)^{n} i \pi}{n} - \frac{\left(-1\right)^{n} + 1}{n^{2}} = 0$$
Solve this equation
Solutions are not found,
function may have no extrema
Inflection points
Let's find the inflection points, we'll need to solve the equation for this
$$\frac{d^{2}}{d n^{2}} f{\left(n \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d n^{2}} f{\left(n \right)} = $$
the second derivative
$$\frac{- \left(-1\right)^{n} \pi^{2} - \frac{2 \left(-1\right)^{n} i \pi}{n} + \frac{2 \left(\left(-1\right)^{n} + 1\right)}{n^{2}}}{n} = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
$$\frac{d^{2}}{d n^{2}} f{\left(n \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d n^{2}} f{\left(n \right)} = $$
the second derivative
$$\frac{- \left(-1\right)^{n} \pi^{2} - \frac{2 \left(-1\right)^{n} i \pi}{n} + \frac{2 \left(\left(-1\right)^{n} + 1\right)}{n^{2}}}{n} = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Vertical asymptotes
Have:
$$n_{1} = 0$$
$$n_{1} = 0$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at n->+oo and n->-oo
Limit on the left could not be calculated
$$\lim_{n \to -\infty}\left(\frac{\left(-1\right)^{n} + 1}{n}\right)$$
Limit on the right could not be calculated
$$\lim_{n \to \infty}\left(\frac{\left(-1\right)^{n} + 1}{n}\right)$$
Limit on the left could not be calculated
$$\lim_{n \to -\infty}\left(\frac{\left(-1\right)^{n} + 1}{n}\right)$$
Limit on the right could not be calculated
$$\lim_{n \to \infty}\left(\frac{\left(-1\right)^{n} + 1}{n}\right)$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of ((-1)^n + 1)/n, divided by n at n->+oo and n ->-oo
Limit on the left could not be calculated
$$\lim_{n \to -\infty}\left(\frac{\left(-1\right)^{n} + 1}{n^{2}}\right)$$
Limit on the right could not be calculated
$$\lim_{n \to \infty}\left(\frac{\left(-1\right)^{n} + 1}{n^{2}}\right)$$
Limit on the left could not be calculated
$$\lim_{n \to -\infty}\left(\frac{\left(-1\right)^{n} + 1}{n^{2}}\right)$$
Limit on the right could not be calculated
$$\lim_{n \to \infty}\left(\frac{\left(-1\right)^{n} + 1}{n^{2}}\right)$$
Even and odd functions
Let's check, whether the function even or odd by using relations f = f(-n) и f = -f(-n).
So, check:
$$\frac{\left(-1\right)^{n} + 1}{n} = - \frac{1 + \left(-1\right)^{- n}}{n}$$
- No
$$\frac{\left(-1\right)^{n} + 1}{n} = \frac{1 + \left(-1\right)^{- n}}{n}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\frac{\left(-1\right)^{n} + 1}{n} = - \frac{1 + \left(-1\right)^{- n}}{n}$$
- No
$$\frac{\left(-1\right)^{n} + 1}{n} = \frac{1 + \left(-1\right)^{- n}}{n}$$
- No
so, the function
not is
neither even, nor odd