3.3b Series labs

Source: Lab ideas from Computational Physics (Landau, Páez, Bordeianu), §3.3.1. Rewritten as original drills. Continues 3.3 Errors with power series.

Eleven checks on the sine recurrence from 3.3: does it converge, to the right answer, for which $x$, and what breaks when you form factorials or ask for more digits than the machine has. Coding stubs fail until filled in; write-ups need the listed keywords. Use math.sin only when the prompt says so — as a reference “exact” value, never as the series engine.

Labs

Activity: Error table

For each $x$ on the input line, sum the sine series with $\varepsilon=10^{-8}$ (recurrence from 3.3). Print one row per $x$:

x N sum rel_err

where rel_err = |sum − math.sin(x)| / |math.sin(x)| (use 1e-30 in the denominator if $\sin x=0$). Format: x with 4 decimals, N as int, sum with 10 significant figures (:.10g), rel_err in scientific notation with 3 digits (:.3e).

Input: space-separated floats. Output: one blank-separated row per $x$, same order.

Write-ups

0/4 solved

3.3b.2

Using “converge”, explain:

For sufficiently small $x$ (e.g. $x\le 1$) with $\varepsilon=10^{-8}$, does the recurrence converge, and does it converge to the correct $\sin x$? One word is enough if it is the right keyword.

Minimum 1 words. Your answer must include: “converge”.

0 words/ 1 min

Meet the writing requirements, then you can open the model answer.

Explorers

Next: 3.4 Specular reflection and Bessel theory.