Test the series for convergence or divergence. ∑from n=1 to ∞ ((n^3 -1)/(n^4+1))
I do the alternating seriestest. It shows that bn >0, (bn+1)<bn and the limit is 0. So, it is convergent but the correct answer is divergent. Why?
why alternating series? All the terms are positive.
lim(n->∞) a_(n+1)/(a_n) = n^3/n^4 = 1/n
The harmonic series diverges
Test the series for convergence or divergence.∑from n=1 to ∞ ((e^n) /n^2)
n = 1
To test the series for convergence or divergence, we can use the Limit Comparison Test.
Step 1: Determine the terms of the series.
The terms of the given series are ((n^3 - 1)/(n^4 + 1)).
Step 2: Find a comparison series.
We need to find a comparison series whose convergence or divergence is already known.
Since we have polynomials of degree 3 and 4, it is useful to compare the given series with a p-series. A p-series has the form ∑(1/n^p), where p is a positive constant.
In this case, let's take p = 4. So, the p-series is ∑(1/n^4).
Step 3: Take the limit of the ratio.
We need to calculate the limit as n approaches infinity of the ratio of the terms of the given series and the p-series.
Let's denote the given series as A(n) = ((n^3 - 1)/(n^4 + 1)) and the p-series as B(n) = 1/n^4.
We can write the limit comparison as:
L = lim(n -> ∞) (A(n) / B(n))
L = lim(n -> ∞) (((n^3 - 1)/(n^4 + 1)) / (1/n^4))
L = lim(n -> ∞) ((n^3 - 1)/(n^4 + 1)) * (n^4/1)
Simplifying further:
L = lim(n -> ∞) (n^7 - n^4) / (n^4 + 1)
Step 4: Evaluate the limit.
To evaluate the limit, we divide the leading terms in the numerator and denominator by the highest power of n.
L = lim(n -> ∞) (n^3 - 1/n^4) / (1 + 1/n^4)
As n approaches infinity, the constants (1/n^4 and 1) become negligible. Therefore, they do not contribute significantly to the limit.
L = lim(n -> ∞) (n^3 / n^4)
L = lim(n -> ∞) (1 / n)
Step 5: Analyze the limit.
Since the limit as n approaches infinity of (1/n) is equal to zero, we can conclude that the limit comparison is zero.
Step 6: Make the conclusion.
According to the Limit Comparison Test, if the limit comparison is a finite positive number, the series and the comparison series either both converge or both diverge.
Since the limit comparison is zero, it means that the given series ((n^3 - 1)/(n^4 + 1)) and the comparison series (∑(1/n^4)) converge or diverge together.
The series ∑((n^3 - 1)/(n^4 + 1)) is a convergent series.
Therefore, the given series converges.