Series converges? or diverges? Which test? explain?
(-1)^n * (4n^3)
------------(fraction)
(n^4) + 1
Since
(-1)^n / n converges, so does this one.
n^3/(n^4+1) < 1/n
To determine whether the series (-1)^n * (4n^3) / (n^4 + 1) converges or diverges, we can use the ratio test.
The ratio test is a common test to check the convergence or divergence of a series. It involves taking the limit as n approaches infinity of the absolute value of the ratio of the (n+1)th term to the nth term of the series.
Let's denote the nth term of the series as a(n) = (-1)^n * (4n^3) / (n^4 + 1).
We will calculate the limit as n approaches infinity of |a(n+1) / a(n)|:
lim(n→∞) |((-1)^(n+1) * (4(n+1)^3))/((n+1)^4 + 1) / (-1)^n * (4n^3) / (n^4 + 1)|
Simplifying the expression, we get:
lim(n→∞) |((-1)^(n+1) * (4(n+1)^3))/(-1)^n * (4n^3)) * ((n^4 + 1))/( (n+1)^4 + 1)|
Dividing the numerator and denominator separately, we obtain:
lim(n→∞) |(-1)^(n+1) * (4(n+1)^3) / (-1)^n * (4n^3)| * |((n^4 + 1))/( (n+1)^4 + 1)|
Simplifying further, we have:
lim(n→∞) |(-1)^(n+1) * (n+1)^3 / n^3| * |((n^4 + 1))/( (n+1)^4 + 1)|
Now, let's evaluate the limit separately for each factor.
1) The limit of |(-1)^(n+1) * (n+1)^3 / n^3| as n approaches infinity:
Taking the absolute value and expanding the numerator, we get:
|(-1)^(n+1) * (n+1)^3 / n^3| = (|(-1)|)^(n+1) * |(n+1)^3 / n^3|
|(-1)^(n+1) * (n+1)^3 / n^3| = (|1|)^(n+1) * (n^3+3n^2+3n+1) / n^3
Simplifying further, we have:
|(-1)^(n+1) * (n+1)^3 / n^3| = n^3+3n^2+3n+1 / n^3
As n approaches infinity, every term except the leading term becomes insignificant. Thus, the limit is 1:
lim(n→∞) n^3+3n^2+3n+1 / n^3 = 1
2) The limit of |((n^4 + 1)) / ((n+1)^4 + 1)| as n approaches infinity:
Expanding the denominator and taking the limit, we get:
lim(n→∞) (n^4 + 1) / (n^4 + 4n^3 + 6n^2 + 4n + 1) = 1
Now, let's multiply the limits of the two factors:
lim(n→∞) |(-1)^(n+1) * (n+1)^3 / n^3| * |((n^4 + 1))/( (n+1)^4 + 1)| = 1 * 1 = 1
According to the ratio test, if the limit is less than 1, the series converges. If the limit is greater than 1 or infinity, the series diverges. In this case, the limit is exactly 1.
Therefore, the ratio test is inconclusive, and we cannot determine whether the series converges or diverges using this specific test.