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Graphing y = ((-1)^n+1)/n

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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           n    
       (-1)  + 1
f(n) = ---------
           n    
f(n)=(1)n+1nf{\left(n \right)} = \frac{\left(-1\right)^{n} + 1}{n}
f = ((-1)^n + 1)/n
The graph of the function
2.03.04.05.06.07.08.09.010.00.02.0
The domain of the function
The points at which the function is not precisely defined:
n1=0n_{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:
(1)n+1n=0\frac{\left(-1\right)^{n} + 1}{n} = 0
Solve this equation
The points of intersection with the axis N:

Analytical solution
n1=1n_{1} = 1
Numerical solution
n1=75n_{1} = -75
n2=27n_{2} = 27
n3=21n_{3} = 21
n4=79n_{4} = -79
The points of intersection with the Y axis coordinate
The graph crosses Y axis when n equals 0:
substitute n = 0 to ((-1)^n + 1)/n.
(1)0+10\frac{\left(-1\right)^{0} + 1}{0}
The result:
f(0)=~f{\left(0 \right)} = \tilde{\infty}
sof doesn't intersect Y
Extrema of the function
In order to find the extrema, we need to solve the equation
ddnf(n)=0\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:
ddnf(n)=\frac{d}{d n} f{\left(n \right)} =
the first derivative
(1)niπn(1)n+1n2=0\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
d2dn2f(n)=0\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:
d2dn2f(n)=\frac{d^{2}}{d n^{2}} f{\left(n \right)} =
the second derivative
(1)nπ22(1)niπn+2((1)n+1)n2n=0\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:
n1=0n_{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
limn((1)n+1n)\lim_{n \to -\infty}\left(\frac{\left(-1\right)^{n} + 1}{n}\right)
Limit on the right could not be calculated
limn((1)n+1n)\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
limn((1)n+1n2)\lim_{n \to -\infty}\left(\frac{\left(-1\right)^{n} + 1}{n^{2}}\right)
Limit on the right could not be calculated
limn((1)n+1n2)\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:
(1)n+1n=1+(1)nn\frac{\left(-1\right)^{n} + 1}{n} = - \frac{1 + \left(-1\right)^{- n}}{n}
- No
(1)n+1n=1+(1)nn\frac{\left(-1\right)^{n} + 1}{n} = \frac{1 + \left(-1\right)^{- n}}{n}
- No
so, the function
not is
neither even, nor odd