Question

Determine with limitrules if following sequence convergence, if yes then calculate its limit
\[a_{n}:=\frac{(-1)^{n}}{n} + \frac{n^{2}+14}{7+n-n^{2}}\] for \[n\geq1\]

i found -1 i am not sure if its true..

Answer 1

i think you do this with your eyeballs right? the first term evidently goes to zero

Answer 2

the second term is a rational function where the degree of the numerator is equal to the degree of the denominator (they are both two) so it has a horizontal asymptote at \(y\) equal the ratio of the leading coefficients, which as you said is \(-1\)

Answer 3

ok thank you satellite, i will write my whole solution to be sure if i done right

Answer 4

your teacher may want you to do something silly like divide top and bottom by \(n^2\) but you probably remember from some pre-calc class that if the degree of the numerator is equal to the degree of the denominator in a rational function, you have a horizontal asymptote at the ratio of the leading coefficients.
this is exactly the same, but instead of \(x\) any number you have \(n\) presumably an integer. it makes no difference

Answer 5

Here is my solution is it looks good so?
\[\Large a_{n}:=\frac{(-1)^{n}}{n} + \frac{n^{2}+14}{7+n-n^{2}}= 0 + \frac{n^{2}(1+\frac{14}{n^{2}})}{n^{2}(\frac{7}{n^{2}}+\frac{1}{n}-1)} =\]
\[\Large
\frac{1+0}{0+0-1}=\frac{1}{-1}=-1\]

Answer 6

dear @satellite73 is my solution right?