Alternating series (Leibniz criterion)

How to use the applet

• $a_n=\frac{1}{(n+1)^p}$, $s_n=\sum_{k=0}^{n}(-1)^k a_k.$
• Move $N$ to show terms and partial sums up to index $N$.
• Move $p$ to change how fast $a_n$ decreases.
• Green: the terms $a_n$.
• Red: the even partial sums $s_{2m}$.
• Blue: the odd partial sums $s_{2m+1}$.
• The dashed horizontal red/blue lines indicate the current bounds $s_{2m+1}\le s \le s_{2m}.$
• The vertical gray segment visualizes the “gap” $s_{2m}-s_{2m+1}=a_{2m+1},$ which tends to $0$ as $m\to\infty$.
• $s_{2m}$ (red) moves downward, $s_{2m+1}$ (blue) moves upward, and the gap shrinks; hence both subsequences converge to the same limit, so $(s_n)$ converges.
• The alternating series $\sum (-1)^k a_k$ converges for every $p>0$ (Leibniz).
• The corresponding positive series $\sum a_k=\sum \frac{1}{(k+1)^p}$ converges only if $p>1$ (and diverges for $0<p\le 1$).

Description

Link to code.

Reference: Lecture Notes Calculus 1 ( May22,2022 ) Theorem3.5 page 36

This applet aims to justify Leibniz criterion.

Move the slider N and watch the partial sums $s_n=\sum_{k=0}^n (-1)^k a_k$.

The even partial sums $s_{2m}$ (red) form a decreasing sequence, while the odd partial sums $s_{2m+1}$ (blue) form an increasing sequence. The dashed horizontal lines visualize the bounds

The vertical “gap” segment shows

so the two bounding subsequences squeeze together and must converge to the same limit. Therefore $(s_n)$ converges and the series converges.

In this applet we use $a_n=\frac{1}{(n+1)^p}$ with $p>0$, which is decreasing and satisfies $a_n\to 0$.

In the code all the sequnce points are computed in advance for performance reasons in computeArrays.

The in the update function each sequence point is set to visible depending on the current N

    aPts[n].setAttribute({ visible: vis });
    sPts[n].setAttribute({ visible: vis });