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   "source": [
    "# Task 02\n",
    "\n",
    "**deadline: 07/03/2021 23:59 CET**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**[0.5 points]:** Write a function that takes $ n \\in \\mathbb{N}_0 $ as an argument and returns the first $ n $ terms of the Maclaurin series of the function $ \\sin{x} $,\n",
    "\n",
    "$$\n",
    "\\sum_{k = 0}^{n} \\frac{(-1)^k}{(2 k + 1)!} x^{2k + 1}\n",
    "$$\n",
    "\n",
    "in the form of [`numpy.poly1d`](https://numpy.org/doc/stable/reference/generated/numpy.poly1d.html) (in order to evaluate the factorial, you may use the function [`scipy.special.factorial`](https://docs.scipy.org/doc/scipy/reference/generated/scipy.special.factorial.html)).\n",
    "\n",
    "**[0.25 points]:** Write a function that calculates the relative error of the function above for arbitrary point $ x \\in \\mathbb{R} $ and $ n \\in \\mathbb{N}_0 $. Print the relative error for $ x = \\pi \\, / \\, 2 $ and $ n = 2$ on 16 decimal places.\n",
    "\n",
    "**[0.25 points]:** Find $ n $ for which the relative error at $ x = \\pi \\, / \\, 2 $ is less than $ 200 \\, \\varepsilon $, where $ \\varepsilon $ is the machine epsilon of the 64-bit representation of IEEE floating point numbers. Print that number."
   ]
  },
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   "execution_count": null,
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   "source": [
    "\n",
    "# add your code here\n"
   ]
  }
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