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The Integral Test for Positive Series Examples 2
Let $f(n) = a_n$ be a continuous ultimately non-increasing positive function for the interval $[N, \infty)$ where $N \in \mathbb{N}$. Recall from The Integral Test for Positive Series page that if $\int_{N}^{\infty} f(x) \: dx$ converges then $\sum_{n=N}^{\infty} a_n$ converges. Similarly, if $\int_{N}^{\infty} f(x) \: dx$ diverges to infinity then $\sum_{n=N}^{\infty} a_n$ diverges to infinity.
We will now look at some examples of applying the integral test.
Example 1
Determine whether the series $\sum_{n=1}^{\infty} \frac{\ln n}{n}$ converges or diverges using the integral test.
Clearly the function $f(x) = \frac{\ln x}{x}$ is continuous on $[1, \infty )$. We can easily verify that for for all $n \in \mathbb{N}$ such that $n > 1$ that $\frac{\ln n}{n} > 0$ since $\ln n > 0$ for $n > 1$ and $n > 0$ for $n ≥ 1$.
Furthermore, it's not hard to see that the sequence $\left \{ \frac{\ln n}{n} \right \}$ is ultimately decreasing since the natural logarithm function grows slower than $n$ ultimately. Therefore, we can use the integral test for this series. We evaluate the following integral:
(1)This integral can be evaluated with substitution. Let $u = \ln x$. Then $du = \frac{1}{x} \: dx$ and so:
(2)Therefore we have that:
(3)By the integral test, we have that since $\int_1^{\infty} \frac{\ln x}{x} \: dx$ diverges, then the series $\sum_{n=1}^{\infty} \frac{\ln n}{n}$ also diverges.
Example 2
Determine whether the series $\sum_{n=2}^{\infty} \frac{1}{n \ln^2 n}$ converges or diverges using the integral test.
Clearly this function is continuous on $[2, \infty)$. For $n \in \mathbb{N}$ and $n ≥ 2$ this series is positive and decreasing, so we can apply the integral test.
We want to thus evaluate the following integral:
(4)We will need to use substitution once again to evaluate this integral. Let $u = \ln x$. Then $du = \frac{1}{x} \: dx$, and so:
(5)Therefore we have that:
(6)Therefore $\int_2^{\infty} \frac{1}{x \ln^2 x} \: dx$ converges and so by the integral test we have that the series $\sum_{n=2}^{\infty} \frac{1}{n \ln^2 n}$ also converges.
Example 3
Reprove the integral test for positive series - that is, prove that if $f(n) = a_n$ where $f$ is a continuous ultimately non-increasing positive function on the interval $[N, \infty)$ where $N \in \mathbb{N}$ then if $\int_N^{\infty} f(x) \: dx$ converges then $\sum_{n=N}^{\infty} a_n$ converges, and if $\int_N^{\infty} f(x) \: dx$ diverges then $\sum_{n=N}^{\infty} a_n$ diverges.
Suppose that $\int_N^{\infty} f(x) \: dx$ converges to $L \in \mathbb{R}$.
Consider the series $\sum_{n=1}^{\infty} a_n$. Let $s_n$ be the $n^{\mathrm{th}}$ partial sum of this series - that is:
(7)Suppose that $n > N$. Then we have that:
(8)The values of $f(N+1)$, $f(N+2)$, …, $f(n)$ represent the areas of rectangles under the function $f$. Therefore:
(9)Since $\int_N^{\infty} f(x) \: dx$ converges to $L$, we have that $0 < s_n ≤ s_N + L$. The sequence $\{ s_n \}$ is an increasing sequence and is bounded above and hence converges which implies that the series $\sum_{n=N}%{\infty} a_n$ converges.
Now suppose that $\int_N^{\infty} f(x) \: dx$ diverges. Suppose that $n > N$. This time we have that:
(10)The values of $f(N)$, $f(N+1)$, …, $f(n)$ overestimate the area of under $f$ and above the positive $x$ axis, and so:
(11)Since the integral $\int_N^{\infty} f(x) \: dx$ diverges to infinity, we must have that by the comparison test that $s_n$ diverges to infinity.