{ "cells": [ { "cell_type": "markdown", "id": "f5bbd897-7d26-4f74-8e2b-37091c944e71", "metadata": {}, "source": [ "# Задачи для практического занятия по курсу:\n", "\n", "## \"ПРОГРАММИРОВАНИЕ\" 👩‍💻\n", "\n", "### Лабораторная работа по теме \"Суммирование числовых рядов\" 📝\n", "\n", "#### Преподаватель практики: Т.Ф. Долгих, каф. ВМ и МФ, мехмат ЮФУ 👩‍🏫\n", "\n", "-----------\n", "\n", "**Переименовать файл:** File -> Rename... -> поменять NameStud на свою фамилию, записанную латинскими буквами -> нажать Rename.\n", "\n", "-----------\n", "\n", "**Общие указания.** \n", "1) Результат каждой суммы печатать с тем количеством знаков после запятой, что указано в задании через точность вычислений.\n", " Например, задана точность вычислений $\\varepsilon = 10^{-n}$. Тогда\n", "```python\n", "print('summa = %(n+2).nf' %s)\n", "```\n", "2) Каждую сумму проверить с помощью `sympy`.\n", "Например, пусть дан числовой ряд вида \n", "$$\n", "\\sum \\limits_{i=\\text{нач}}^{\\infty} a_i\n", "$$\n", "Тогда проверка будет иметь вид\n", "```python\n", "from sympy import Symbol, Sum, oo\n", "\n", "i = Symbol('i')\n", "ss = Sum(ai, (i, нач, oo)).evalf()\n", "\n", "print('summa sympy = %(n+2).nf' %ss) # установить нужный формат\n", "```\n", "\n", "-----------\n", "\n", "Для визуализации формул можно использовать сайт https://www.processon.io/ru/visual \n", "\n", "-----------\n", "\n", "📜 **Пример 1.**\n", "Вычислить приближенное значение бесконечной суммы\n", "$$\n", "1 - \\frac{1}{2^2} + \\frac{1}{3^2} - \\frac{1}{4^2} + \\ldots\n", "$$\n", "Вычисления продолжать до тех пор пока очередное слагаемое по абсолютной величине не окажется меньше $\\varepsilon = 3 \\cdot 10^{-4}$." ] }, { "attachments": { "ff0d1bcf-139a-412a-b68f-01d650023427.png": { "image/png": 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" } }, "cell_type": "markdown", "id": "ee296401-3f9c-4aba-8938-4a53a3c4033b", "metadata": {}, "source": [ "**Решение.** Для решения воспользуемся общей формулой вида\n", "\n", "![image.png](attachment:ff0d1bcf-139a-412a-b68f-01d650023427.png)" ] }, { "cell_type": "code", "execution_count": 1, "id": "ae3603b7-cbcb-4919-af6b-5de34d511392", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "summa = 0.8226\n" ] } ], "source": [ "# программа \n", "\n", "eps = 3e-4\n", "\n", "s = 0 # для БЕСКОНЕЧНОЙ суммы!!!\n", "i = 0\n", "z = 1\n", "a = 1\n", "while abs(a) >= eps:\n", " s += a\n", " i += 1 # только +1\n", " z = -z\n", " a = z / (i + 1) ** 2\n", "\n", "print('summa = %6.4f' %s)" ] }, { "attachments": {}, "cell_type": "markdown", "id": "7bf8600a-d6d6-4d8e-b2a1-a32e624c6328", "metadata": {}, "source": [ "💻 **Задание 1.**\n", "Вычислить приближённое значение бесконечной суммы\n", "$$\n", "\\frac{1}{2 \\cdot 3} + \\frac{1}{3 \\cdot 4} + \\frac{1}{4 \\cdot 5} + \\frac{1}{5 \\cdot 6} + \\ldots\n", "$$\n", "Вычисления продолжать до тех пор пока очередное слагаемое не окажется меньше $\\varepsilon = 10^{-6}$." ] }, { "attachments": {}, "cell_type": "markdown", "id": "08328a36-155a-45b3-b8ce-636e52ff08f6", "metadata": {}, "source": [ "**Решение.** Для решения воспользуемся общей формулой вида\n" ] }, { "cell_type": "code", "execution_count": null, "id": "3a8ab3c5-8a64-480e-ae04-09d38296767d", "metadata": {}, "outputs": [], "source": [ "# программа " ] }, { "cell_type": "code", "execution_count": null, "id": "74e5313a-12fa-4534-86df-d6098e82c3ab", "metadata": {}, "outputs": [], "source": [ "# проверка" ] }, { "cell_type": "markdown", "id": "97d2b9f4-995b-45df-a8f5-0f4751f12a65", "metadata": {}, "source": [ "💻 **Задание 2.**\n", "Вычислить приближённое значение бесконечной суммы\n", "$$\n", "\\frac{1}{1 \\cdot \\sqrt{2}} + \\frac{1}{2 \\cdot \\sqrt{3}} + \\frac{1}{3 \\cdot \\sqrt{4}} + \\frac{1}{4 \\cdot \\sqrt{5}} + \\ldots\n", "$$\n", "Вычисления продолжать до тех пор пока очередное слагаемое не окажется меньше $\\varepsilon = 10^{-5}$." ] }, { "cell_type": "markdown", "id": "04056ea4-805b-40f2-b458-ff2effd9e212", "metadata": {}, "source": [ "**Решение.** Для решения воспользуемся общей формулой вида\n" ] }, { "cell_type": "code", "execution_count": null, "id": "e943420c-eb40-4aa8-9b59-12555d2578d6", "metadata": {}, "outputs": [], "source": [ "# программа" ] }, { "cell_type": "code", "execution_count": null, "id": "2564471b-690d-48de-b17f-8f0dd190cfc3", "metadata": {}, "outputs": [], "source": [ "# проверка" ] }, { "cell_type": "markdown", "id": "2a69d29c-8c15-4d9b-8063-69652d53fa44", "metadata": {}, "source": [ "💻 **Задание 3.**\n", "Вычислить приближённое значение бесконечной суммы\n", "$$\n", "1 - \\frac{1}{3^3} + \\frac{1}{5^3} - \\frac{1}{7^3} + \\ldots\n", "$$\n", "Вычисления продолжать до тех пор пока очередное слагаемое по абсолютной величине не окажется меньше $\\varepsilon = 10^{-4}$." ] }, { "cell_type": "markdown", "id": "a810ce2b-f2a4-44b7-aa67-2e27583a3728", "metadata": {}, "source": [ "**Решение.** Для решения воспользуемся общей формулой вида\n" ] }, { "cell_type": "code", "execution_count": null, "id": "a0774f6a-5fd1-4e01-9c4c-4c2355664d86", "metadata": {}, "outputs": [], "source": [ "# программа" ] }, { "cell_type": "code", "execution_count": null, "id": "95d5ea17-fc35-4338-a93b-ed0cebdbba33", "metadata": {}, "outputs": [], "source": [ "# проверка" ] }, { "attachments": {}, "cell_type": "markdown", "id": "f3bf644b-ca66-4d5a-951f-5aa8ab5427c2", "metadata": {}, "source": [ "💻 **Задание 4.**\n", "Вычислить приближённое значение бесконечной суммы\n", "$$\n", "1 - \\frac{1}{2} + \\frac{1}{5} - \\frac{1}{10} + \\ldots\n", "$$\n", "Вычисления продолжать до тех пор пока очередное слагаемое по абсолютной величине не окажется меньше $\\varepsilon = 4 \\cdot 10^{-5}$." ] }, { "attachments": {}, "cell_type": "markdown", "id": "08cb6457-8c75-4794-b3a4-a05d5be1c73d", "metadata": {}, "source": [ "**Решение.** Для решения воспользуемся общей формулой вида\n" ] }, { "cell_type": "code", "execution_count": null, "id": "1c21c4ce-d987-4062-9820-5b0382747435", "metadata": {}, "outputs": [], "source": [ "# программа" ] }, { "cell_type": "code", "execution_count": null, "id": "b7128e5b-918e-489b-a248-5c76124f172b", "metadata": {}, "outputs": [], "source": [ "# проверка" ] } ], "metadata": { "kernelspec": { "display_name": "Python 3 (ipykernel)", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.13.9" } }, "nbformat": 4, "nbformat_minor": 5 }