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- https://math.libretexts.org/Bookshelves/Analysis/Real_Analysis_(Boman_and_Rogers)/10%3A_Epilogue_to_Real_AnalysisThumbnail: Real number line with some constants such as \(\pi\). (Public Domain; User:Phrood).
- https://math.libretexts.org/Bookshelves/Analysis/Real_Analysis_(Boman_and_Rogers)/04%3A_Convergence_of_Sequences_and_SeriesThumbnail: Leonhard Euler. (Public Domain; Jakob Emanuel Handmann).
- https://math.libretexts.org/Bookshelves/Analysis/Real_Analysis_(Boman_and_Rogers)/03%3A_Questions_Concerning_Power_Series/3.E%3A_Questions_Concerning_Power_Series_(Exercises)Use Taylor’s formula to find the Taylor series of the given function expanded about the given point \(a\). \(f(x) = \ln(1 + x),\; a = 0\) \(f(x) = e^x,\; a = -1\) \(f(x) = x^3 + x^2 + x + 1,\; a = 0\)...Use Taylor’s formula to find the Taylor series of the given function expanded about the given point \(a\). \(f(x) = \ln(1 + x),\; a = 0\) \(f(x) = e^x,\; a = -1\) \(f(x) = x^3 + x^2 + x + 1,\; a = 0\) \(f(x) = x^3 + x^2 + x + 1,\; a = 1\)
- https://math.libretexts.org/Bookshelves/Analysis/Real_Analysis_(Boman_and_Rogers)/10%3A_Epilogue_to_Real_Analysis/10.01%3A_On_the_Nature_of_NumbersThree friends meet in a garden for lunch in Renassaince Italy. Prior to their meal they discuss the book How We Got From There to Here: A Story of Real Analysis. How they obtained a copy is not clear.
- https://math.libretexts.org/Bookshelves/Analysis/Real_Analysis_(Boman_and_Rogers)/05%3A_Convergence_of_the_Taylor_Series-_A_Tayl_of_Three_RemaindersThumbnail: Brook Taylor (1685-1731) was an English mathematician who is best known for Taylor's theorem and the Taylor series.
- https://math.libretexts.org/Bookshelves/Analysis/Real_Analysis_(Boman_and_Rogers)/06%3A_Continuity_-_What_It_Isnt_and_What_It_Is/6.E%3A_Continuity_-_What_It_Isn%E2%80%99t_and_What_It_Is_(Exercises)Use the definition of continuity to prove that \(\ln x\) is continuous at \(1\). [Hint: You may want to use the fact \(\left |\ln x \right | < \varepsilon \Leftrightarrow -\varepsilon < \ln x < \varep...Use the definition of continuity to prove that \(\ln x\) is continuous at \(1\). [Hint: You may want to use the fact \(\left |\ln x \right | < \varepsilon \Leftrightarrow -\varepsilon < \ln x < \varepsilon\) to find a \(δ\).] Write a formal definition of the statement \(f\) is not continuous at \(a\), and use it to prove that the function \(f(x) = \begin{cases} x & \text{ if } x\neq 1 \\ 0 & \text{ if } x= 1 \end{cases}\) is not continuous at \(a = 1\).
- https://math.libretexts.org/Bookshelves/Analysis/Real_Analysis_(Boman_and_Rogers)/05%3A_Convergence_of_the_Taylor_Series-_A_Tayl_of_Three_Remainders/5.03%3A_Cauchy%E2%80%99s_Form_of_the_RemainderIn his 1823 work, Résumé des le¸cons données à l’ecole royale polytechnique sur le calcul infintésimal, Augustin Cauchy provided another form of the remainder for Taylor series.
- https://math.libretexts.org/Bookshelves/Analysis/Real_Analysis_(Boman_and_Rogers)/07%3A_Intermediate_and_Extreme_Values/7.04%3A_The_Supremum_and_the_Extreme_Value_TheoremA continuous function on a closed, bounded interval must be bounded. Boundedness, in and of itself, does not ensure the existence of a maximum or minimum. We must also have a closed, bounded interval.
- https://math.libretexts.org/Bookshelves/Analysis/Real_Analysis_(Boman_and_Rogers)/08%3A_Back_to_Power_Series/8.01%3A_Uniform_ConvergenceWe will now draw our attention back to the question that originally motivated these definitions, “Why are Taylor series well behaved, but Fourier series are not necessarily?” More precisely, we mention...We will now draw our attention back to the question that originally motivated these definitions, “Why are Taylor series well behaved, but Fourier series are not necessarily?” More precisely, we mentioned that whenever a power series converges then whatever it converged to was continuous. Moreover, if we differentiate or integrate these series term by term then the resulting series will converge to the derivative or integral of the original series. This was not always the case for Fourier series.
- https://math.libretexts.org/Bookshelves/Analysis/Real_Analysis_(Boman_and_Rogers)/07%3A_Intermediate_and_Extreme_Values/7.02%3A_Proof_of_the_Intermediate_Value_TheoremThe Intermediate Value Theorem states that if a continuous function, f, with an interval, [a, b], as its domain, takes values f(a) and f(b) at each end of the interval, then it also takes any value be...The Intermediate Value Theorem states that if a continuous function, f, with an interval, [a, b], as its domain, takes values f(a) and f(b) at each end of the interval, then it also takes any value between f(a) and f(b) at some point within the interval. We now have all of the tools to prove the Intermediate Value Theorem.
- https://math.libretexts.org/Bookshelves/Analysis/Real_Analysis_(Boman_and_Rogers)/08%3A_Back_to_Power_Series/8.04%3A_Boundary_Issues_and_Abel%E2%80%99s_TheoremThe integrations we performed in Chapter 2 are legitimate due to the Abel's theorem which extends uniform convergence to the endpoints of the interval of convergence even if the convergence at an endp...The integrations we performed in Chapter 2 are legitimate due to the Abel's theorem which extends uniform convergence to the endpoints of the interval of convergence even if the convergence at an endpoint is only conditional. Abel did not use the term uniform convergence, as it hadn’t been defined yet, but the ideas involved are his.
