M248QuizWeek3_2.ipynb

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    "## 1) List TEN things you learned during Weeks 1 and 2 and FIVE things you still don't understand or are curious and would like to learn about."
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    "# Fibonacci Numbers\n",
    "- What are Fibonacci numbers?\n",
    "- Give examples supporting the statement *Fibonacci numbers are abundant in nature*. \n",
    "- Write a program that outputs a vector containing the first 40 Fibonacci numbers $$F_{n+2}=F_n+F_{n+1}, F_1=0, F_2=1.$$\n",
    "- What is the Golden ratio?\n",
    "- What's the relationship between the Golden Ratio and Fibonacci numbers.\n",
    "- Write a program that outputs a sequence converging to the Golden Ratio.\n",
    "- Is there an explicit formula for the Fibonacci sequence? ($a_n=f(n)$)"
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    "# Consider the series \n",
    "$$\\sum_{n=1}^\\infty\\frac{1}{\\sqrt{n}}$$\n",
    "1. Is the above series convergent or divergent? Explain.\n",
    "2. Use an integral bound in order to quantify how fast the series converges/diverges in terms of $n$.\n",
    "3. Write a program that outputs the number of terms that you have to add up from the series in order to exceed the number $20$. Comment on the speed of convergence/divergence of the series."
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    "# Prove analytically that the series $\\sum_{n=1}^\\infty\\frac{1}{n^6}=\\frac{\\pi^6}{945}$ converges like a power law and find the power (hint: use an integral error bound.) Programming: Verify your result numerically (make a log-log plot for the error sequence and find the slope of the line. Does it match your analytical result?)."
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    "# Alternating series: Decide analytically how many terms of the series $\\sum_{n=1}^\\infty \\frac{(-1)^{n+1}}{\\sqrt{n}}$ will have to be used to be sure the estimate to the sum is accurate to within $0.005$."
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    "# Alternating series: Compute analytically how many terms you would need to use to calculate $\\sum_{n=1}^\\infty\\frac{(-1)^{n+1}}{n^2}$ to within $0.0005$ accuracy. Do you know the limit of this series? Does this series converge like a power law, or some other law? If yes, what is the power?"
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