Convergent terms, divergent series

Necessary condition (Theorem 3.2):

If $\sum_{k=0}^{\infty} a_k$ converges, then $a_n \to 0$ . Equivalently, if $a_n \not\to 0$ , then the series cannot converge.

But a_n → 0 is not sufficient:

The condition $a_n\to 0$ does not guarantee convergence of $\sum a_n$ . This applet provides a clear counterexample using a p-series with $p<1$ .

How to use the applet

Sequence and partial sums used: $a_n=\frac{1}{(n+1)^{1/3}}$ , $s_n=\sum_{k=0}^{n} a_k.$
Move the slider $N$ to display the values up to index $N$ .
Blue points represent the terms $a_n$ .
Red points represent the partial sums $s_n$ .
The larger blue/red markers indicate the current values $a_N$ and $s_N$ .
As $N$ increases, the blue points approach $0$ : $a_n\to 0$ .
Nevertheless, the red points keep increasing and do not approach a finite limit: $(s_n)$ diverges, so the series $\sum_{k=0}^\infty a_k$ diverges.

Loading graph…

Hold Shift + scroll to zoom
Hold Shift + drag to move
Points shaped like <> act as sliders

Description

Link to code.

Reference: Lecture Notes Calculus 1 ( May22,2022 ) Example3.2 ii) page 35

This applet shows an example (similar to the harmonic serie) where the terms converge to $0$ , but the series diverges:

a_n=\frac{1}{(n+1)^{1/3}}, \qquad s_n=\sum_{k=0}^{n}\frac{1}{(k+1)^{1/3}}.

Blue points represent the terms $a_n$ . Red points represent the partial sums $s_n$ .

const termPts: JXG.Point[] = [];
const sumPts: JXG.Point[] = [];

for (let n = 0; n <= MAX_N; n++) {
    termPts.push(
        board.create("point", [n, A[n]], {
            ...})
    );

    sumPts.push(
        board.create("point", [n, S[n]], {
            ...})
    );
}

Move the slider N and observe that the blue values approach the dashed line $y=0$ (so $a_n\to 0$ ), while the red values keep increasing (so $(s_n)$ does not converge and the series diverges). This is a $p$ -series with $p=\tfrac13<1$ , so $\sum (n+1)^{-p}$ diverges even though $a_n\to 0$ .

To highlight the trend, the applet also draws two continuous “trajectory” curves using functiongraph. The first is a continuous model for the terms,

f(x)=\frac{1}{(x+1)^{1/3}}, \qquad \text{so } f(n)=a_n.

The second is a growth model for the partial sums based on integral comparison,

g(x)=\int_0^x \frac{1}{(t+1)^{1/3}}\,dt = \frac{3}{2}\big((x+1)^{2/3}-1\big),

and since $g(x)\to\infty$ , it indicates that the partial sums grow without bound.

In the code these trajectories are created as:

board.create("functiongraph", [(x) => 1 / Math.cbrt(x + 1), 0, MAX_N], ...);
board.create("functiongraph", [(x) => 1.5 * (Math.pow(x + 1, 2/3) - 1), 0, MAX_N], ...);