Why is $\int_1^{\infty}(1/x^2)\,dx$ less than $\sum_{n=1}^{\infty} 1/(n^2)$?

The value of $\displaystyle\int_1^{\infty}\frac{1}{x^2}\,dx$ equals 1.

The series $\displaystyle \sum_{n=1}^{\infty}\frac{1}{n^2}$ equals $\pi^2/6 \approx 1.66493$.

Shouldn't the area under the curve of the function be larger than the sum of the terms of the sequence?

Solution 1:

Not if the rectangles (i.e. the terms in the series) stick out above the curve we're integrating:

(see the page on the integral test)

Solution 2:

Why? Because the function $x\mapsto1/x^2$ to be integrated is nonincreasing. For the same reason, the integral $I$ is greater than the series $S$ minus its first term: $S-1<I<S$ (as a matter of fact, $S=1.66493$ and $I=1$).

Solution 3:

I will write down a brief "lemma" so to say which I've learned from Piskunov's Calculus. Which generalizes this:

Let $y=f(x)$ be a function. Let $a_n = f(n)$ and $s_n = \displaystyle \sum_{k=1}^n a_n$. Then

$$\int_1^{n+1} f(x) dx < s_{n+1} < a_1+\int_1^{n+1}f(x)dx$$

Then $s_n$ converges only if the integral converges, and

$$\int_1^{\infty} f(x) dx < s < a_1+\int_1^{\infty}f(x)dx$$

In your case you have $a_1 = 1$, thus you have $1 < s < 2$, but the improper integral is smaller than the sum, as you can see from the lemma.

This can be derived by simply inspectioning the partial sums as rectangles of height $f(k)$ and width $1$, and comparing the upper and lower sums and the actual integral to the value of the series. Some graphing is all it takes.