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17Calculus Infinite Series - Practice Problem 2441

Infinite Series Practice Problem 2441

\(\displaystyle{ \sum_{n=1}^{\infty}{\frac{n^2-1}{n^3+4}}}\)

Determine the convergence or divergence of the series.
If the series converges,
- determine the value the series converges to, if possible; and
- determine if the series converges absolutely or conditionally.

Choosing Which Tests To Try

Rejecting the Obvious

1. We can tell right away that this is not an alternating series, telescoping series or one of the other special series. So we will not be able to determine what the series converges to.
2. We can tell that the limit of \(a_n\) is zero. So the divergence test will not help us.
3. The integral is quite complicated, so we will not try the integral test unless all the other tests fail.
4. There are no factors with n in the powers. So we will put the root test aside and come back to it only if the other tests fail. We would probably try this one before the integral test.

What We Are Left With

5. We are left with the ratio test and the comparison tests.

Applying The Tests

Ratio Test

Limit Comparison Test

Use the test series \(\sum{1/n}\), a divergent p-series.
\(\begin{array}{rcl} \displaystyle{ \lim_{n\to\infty}{ \left| \frac{a_n}{t_n} \right| } } & = & \displaystyle{ \lim_{n\to\infty}{ \left[ \frac{n^2-1}{n^3+4} \frac{n}{1} \right] } } \\ & = & \displaystyle{ \lim_{n\to\infty}{ \left[ \frac{n^3-n}{n^3+4} \frac{1/n^3}{1/n^3} \right] } } \\ & = & \displaystyle{ \lim_{n\to\infty}{ \left[ \frac{1-1/n^2}{1+4/n^3} \right] } = 1} \\ \end{array}\)
Since the limit is finite and positive, both series either converge or diverge. Since the test series diverges, so does the original series by the limit comparison test.

Direct Comparison Test

For this question \(\displaystyle{ a_n = \frac{n^2-1}{n^3+4} }\). If we think about what happens when n is very large, the \(n^2\) term dominates the 1 in the numerator and the \(n^3\) term dominates the 4 in the denominator. When that happens we are very close to \(n^2/n^3 = 1/n\). So let's apply the direct comparision test using \(t_n = 1/n\). Since \(\sum{1/n}\) is a divergent p-series, we will assume the series \(\sum{a_n}\) diverges also. Using the assumption, the inequality we need to prove is \( 0 \lt t_n \leq a_n \).
It is obvious that \(0 \lt t_n\), so we will focus on showing that \(t_n \leq a_n\).
\(\begin{array}{rcl} t_n & \leq & a_n \\ \displaystyle{ \frac{1}{n} } & \leq & \displaystyle{ \frac{n^2-1}{n^3+4} } \\ n^3+4 & \leq & n(n^2-1) \\ n^3+4 & \leq & n^3-n \\ 4 & \leq & -n \\ -4 & \geq & n \end{array}\)
This last inequality does not hold. So we need another test series.
At this point, we will still assume that the series diverges. So we need a divergent series to compare it to. To find one, we plotted the functions \(f(x)=(x^2-1)/(x^3+4)\) and \(g(x)=1/x\) on the same set of axes. Sure enough, g(x) is larger than f(x) which confirms our work above. Our test series needs to be less than or equal to our original series. So we need a divergent series that is smaller than \(1/n\).
To make \(1/n\) smaller, we can make the denominator larger, so we looked at \(1/(n+1)\). Plotting \(h(x)=1/(x+1)\) on the same set of axes as f(x) and g(x), we can zoom in and see that, sure enough, h(x) is smaller than f(x). But is \(\sum{1/(n+1)}\) still a divergent series? We can use the limit comparison test as follows to find out.
\(\begin{array}{rcl} \displaystyle{ \lim_{n\to\infty}{\frac{t_n}{a_n}} } & = & \displaystyle{ \lim_{n\to\infty}{ \frac{1}{n+1} \frac{n}{1} } } \\ & = & \displaystyle{ \lim_{n\to\infty}{ \frac{n}{n+1} } } \\ & = & \displaystyle{ \lim_{n\to\infty}{ \frac{1}{1+1/n} } = 1 } \end{array}\)
Since the limit is finite and positive, then both series either converge or diverge. Since the test series diverges, so does \(\sum{1/(n+1)}\).
So now for the direct comparison test, we have a divergent test series \(t_n=1/(n+1)\) and we need to show that \( 0 \lt t_n \leq a_n \). As before \(0 \lt t_n\), so we focus on \(t_n \leq a_n\).
\(\begin{array}{rcl} t_n & \leq & a_n \\ \displaystyle{ \frac{1}{n+1} } & \leq & \displaystyle{ \frac{n^2-1}{n^3+4} } \\ n^3+4 & \leq & (n+1)(n^2-1) \\ n^3+4 & \leq & n^3-n+n^2-1 \\ 4 & \leq & n^2-n-1 \\ 0 & \geq & n^2-n-5 \end{array}\)
Can we find an N such that for \(n \gt N\), \(n^2-n-5 \geq 0\)? Yes we can.
Notice that \(y=x^2-x-5\) is a parabola opening up. So we need to find where the graph crosses the x-axis and choose N larger than that. Completing the square gives us \(x=1/2\pm\sqrt{21/4}\). We don't care about the negative value. The positive value is about 2.79. So we choose N=3. So the inequality \( 0 \geq n^2-n-5 \) holds for \( n \gt 3 \) and therefore the series diverges.

Test Summary List and Answer

test/series

works?

notes

divergence test

no

limit is zero, so test is inconclusive

p-series

no

not a p-series

geometric series

no

not a geometric series

alternating series test

no

not an alternating series

telescoping series

no

not a telescoping series

ratio test

no

inconclusive

limit comparison test

yes

best test

direct comparison test

yes

integral test

no

integral too complex

root test

Final Answer

The series \(\displaystyle{ \sum_{n=1}^{\infty}{\frac{n^2-1}{n^3+4}}}\) diverges.

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Topics You Need To Understand For This Page

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