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Real Analysis 1-4 (semesters 1-4)
    Basic notions. Axioms of the real numbers (semester 1, weeks 1-2; 0 problems)
        Fundaments of Logic (semester 1, week 1; 16 problems)
        Proving Techniques: Proof by Contradiction, Induction (semester 1, week 1; 20 problems)
            Fibonacci Numbers (semester 1, week 1; 6 problems)
        Solving Inequalities and Optimization Problems by Inequalities between Means (semester 1, week 1; 19 problems)
        Sets, Functions, Combinatorics (semester 1, week 2; 22 problems)
    Axioms of the real numbers (semester 1, weeks 2-4; 0 problems)
        Field Axioms (semester 1, week 2; 5 problems)
        Ordering Axioms (semester 1, week 2; 6 problems)
        The Archimedean Axiom (semester 1, week 2; 3 problems)
        Cantor Axiom (semester 1, week 2; 7 problems)
        The Real Line, Intervals (semester 1, week 3; 15 problems)
        Completeness Theorem, Connectivity, Topology of the Real Line. (semester 1, week 3; 4 problems)
        Powers (semester 1, week 4; 3 problems)
    Convergence of Sequences (semester 1, weeks 4-6; 0 problems)
        Theoretical Exercises (semester 1, week 4; 61 problems)
        Order of Sequences, Threshold Index (semester 1, week 4; 22 problems)
        Limit Points, liminf, limsup (semester 1, week 5; 15 problems)
        Calculating the Limit of Sequences (semester 1, week 5; 33 problems)
Problem 277

     (1)  \(\displaystyle \lim {(-1)^n\over n}=?\)      (2)  \(\displaystyle \lim {1\over n!}=?\)

    Difficulty: 1.


Problem 278

    Guess the limit, and prove using the definition:

    \(\displaystyle \lim {n\over 2^n}=?\)

    Difficulty: 2.


Problem 279

    Determine the limit of \(\displaystyle \displaystyle{\frac{n^2 +1}{n+1} -an}\) for all values of \(\displaystyle a\).

    Difficulty: 2.


Problem 280

    Determine the limit of \(\displaystyle \sqrt{n^2 -n +1} -an\) for all values of \(\displaystyle a\).

    Difficulty: 3.


Problem 281

    Prove that \(\displaystyle \sqrt[n]{2}\to 1\).

    Difficulty: 3.


Problem 284

    \(\displaystyle \lim {n^2+6n^3-2n+10 \over -4n-9n^3+10^{10}}=?\)

    Difficulty: 3.


Problem 285

    \(\displaystyle \lim {n+7\sqrt{n} \over 2n\sqrt{n}+3}=?\)

    Difficulty: 3.


Problem 282

    Calculate \(\displaystyle \lim_{n\to \infty} \sqrt[n]{2^n-n}.\)

    Difficulty: 4. Solution is available for this problem.


Problem 283

    Guess the limits, and prove using the definition:

    \(\displaystyle \lim {2^n\over n!}=?\)

    Difficulty: 4.


Problem 286

    Calculate the following:

    \(\displaystyle \lim {n^{100} \over 1,1^n}=?\)

    Difficulty: 4.


Problem 288

    Calculate the limit of the sequence \(\displaystyle \sqrt[n]{n!}\).

    Difficulty: 4.


Problem 289

    Calculate the limit of the following sequences.

     (1)  \(\displaystyle \frac{n^{5}-n^{3}+1}{{3n^5-2n^4+8}}\);      (2)  \(\displaystyle \sqrt {n^4+n^2} -n^2\);      (3)  \(\displaystyle \sqrt[n]{6^n-5^n}\).    

    Difficulty: 4.


Problem 290

     (1)  \(\displaystyle \root{n}\of{3}\)      (2)  \(\displaystyle \root{n}\of{\frac{1}{n}}\)      (3)  \(\displaystyle \ds\left(\frac{1+\log2}{n}\right)^n\)      (4)  \(\displaystyle \root{n}\of{2^n+n}\)

    Difficulty: 4.


Problem 291

    Calculate the limit of the following sequences.

     (1)  \(\displaystyle \frac{3n+16}{4n-25}\),      (2)  \(\displaystyle n\cdot\left( \sqrt {1+\frac{1}{n}}-1 \right)\),      (3)  \(\displaystyle \frac{1}{n}\cdot \frac{n^2+1}{n^3+1}\),      (4)  \(\displaystyle \frac{5-2n^2}{4+n}\),
     (5)  \(\displaystyle \frac{\sin (n)+n}{n}\),      (6)  \(\displaystyle \frac{2n^3+3\sqrt n }{1-n^3}\),      (7)  \(\displaystyle \sqrt[n]{n+5^n}\),      (8)  \(\displaystyle \frac{2^n+n!}{n^n-n^{1000}}\),
     (9)  \(\displaystyle \sqrt[n]{n^n-5^n}\),      (10)  \(\displaystyle \frac{\sin(n)}{n} \),      (11)  \(\displaystyle \frac{5n^2+(-1)^n}{8n}\),      (12)  \(\displaystyle \frac{6n+2n^2\cdot (-1)^n}{n^2}\).

    Difficulty: 4.


Problem 292

    Calculate the limit of the following sequences.

     (1)  \(\displaystyle \sqrt[n]{2n+\sqrt n}\),      (2)  \(\displaystyle \frac{n^7-6n^6+5n^5-n-1}{n^3+n^2+n+1}\),      (3)  \(\displaystyle \frac{n^3+n^2\sqrt n -\sqrt n +1}{2n^3-6n+\sqrt n -2}\),      (4)  \(\displaystyle \sqrt[n]{\frac{1}{n}-\frac{2}{n^2}}\),
     (5)  \(\displaystyle \sqrt[n]{2^n+3^n}\),      (6)  \(\displaystyle \frac{\sqrt{2n+1} }{\sqrt {3n} +4}\),      (7)  \(\displaystyle \log \frac{n+1}{n+2}\),      (8)  \(\displaystyle \frac{7^n-7^{-n}}{7^n+7^{-n}}\),
     (9)  \(\displaystyle \frac{(2n+3)^5\cdot (18n+17)^{15}}{(6n+5)^{20}}\),      (10)  \(\displaystyle \frac{\sqrt{4n^2+2n+100}}{\sqrt[3]{6n^3-7n^2+2}}\),     (11)  \(\displaystyle \frac{\sqrt[4]{n^3+6}}{\sqrt[3]{n^2+3n-2}}\),
     (12)  \(\displaystyle n\cdot(\sqrt {n+1} -\sqrt n )\),      (13)  \(\displaystyle \frac{2^n+5^n}{3^n+1}\),      (14)  \(\displaystyle n\cdot(\sqrt {n^2+n} -\sqrt {n^2-n})\).

    Difficulty: 4.


Problem 293

    \(\displaystyle \lim {1\over n(\sqrt{n^2-1}-n)}=?\)

    Difficulty: 4. Solution is available for this problem.


Problem 294

    \(\displaystyle \lim \left( {4n+1\over 4n+8}\right)^{3n+2}=?\)

    Difficulty: 4.


Problem 295

    Let \(\displaystyle a>0\).

    \(\displaystyle \lim {}\sqrt[n]{n+a^n}=?\)

    Difficulty: 4.


Problem 297

    \(\displaystyle \lim {1-2+3-4+\ldots-2n \over 2n+1}=?\)

    Difficulty: 4.


Problem 299

    Calculate the following:

    \(\displaystyle \lim\left( \sqrt{2}\cdot\sqrt[4]{2}\cdot \sqrt[8]{2}\cdot\ldots\cdot\sqrt[2^n]{2}\right) =? \)

    Difficulty: 4.


Problem 300

    Is it convergent?

    \(\displaystyle \sqrt[n]{n^2+\cos n} \)

    Difficulty: 4. Solution is available for this problem.


Problem 301

    Calculate the following

    \(\displaystyle \lim \sqrt[n]{2^n+\sin n}.\)

    Difficulty: 4.


Problem 303

    Calculate the limit of the following sequences.

        (1)  \(\displaystyle \frac{6n^4+2n^2\cdot (-1)^n}{n^4}\),      (2)  \(\displaystyle \sqrt{n^2 +2} +\sqrt{n^2 -2} -2n;\)
        (3)  \(\displaystyle \frac{\sqrt[n]{n^n-5^n}}{n}\),      (4)  \(\displaystyle n\cdot(\sqrt {n^2+n} -\sqrt {n^2-n})\).

    Difficulty: 4.


Problem 306

    Let \(\displaystyle |a|, |b|<1\).

    \(\displaystyle \lim {1+a+a^2+\ldots+a^n \over 1+b+b^2+\ldots+b^n} =?\)

    Difficulty: 4.


Problem 307

    Calculate:
     (1)  \(\displaystyle \lim\root{n^2}\of{1^2+2^n+3^n+\ldots+n^n}=? \)
     (2)  \(\displaystyle \lim\sqrt[n]{\displaystyle 1+\frac12+\frac13+\ldots+\frac1n} = ?\)
     (3)  \(\displaystyle \lim\frac{\displaystyle\frac1{n^2}+\frac1{(n+2)^3}}{ \displaystyle\frac1{n!}-\frac1{\sqrt{(n^2+1)(n^4+2)}}}=?\)

    Difficulty: 4.


Problem 308

    \(\displaystyle \lim\dfrac{1+\dfrac1{\sqrt{2}}+\dfrac1{\sqrt{3}}+\ldots+\dfrac1{\sqrt{n}}}{\sqrt{n}} = ? \)

    Difficulty: 4.


Problem 309

    \(\displaystyle \lim_{n\to\infty} \sqrt[n]{1+\sqrt2+\sqrt[3]{3}+\ldots+\sqrt[n]{n}} = ? \)

    Difficulty: 4.


Problem 287

    Calculate the limit of the sequence \(\displaystyle \sqrt[n]{n}\).

    Difficulty: 5.


Problem 298

    Is it convergent

    \(\displaystyle x_n={\sin 1\over 2}+{\sin 2\over 2^2}+\ldots +{\sin n \over 2^n}?\)

    Difficulty: 5.


Problem 302

    Calculate the following

    \(\displaystyle \lim { {}\sqrt[n]{n!} \over n}.\)

    Difficulty: 5.


Problem 304

    Suppose that \(\displaystyle a_1,a_2,\ldots, a_k>0\). Calculate the limit of the sequence \(\displaystyle \sqrt[n]{a_1^n+a_2^n+\ldots+a_k^n}\).

    Difficulty: 5.


Problem 305

    Calculate the limit of the sequence \(\displaystyle \left(\sqrt{n+\sqrt{n+\sqrt{n}}}-\sqrt{n}\right)\).

    Difficulty: 5.


Problem 296

    Is the sequence

    \(\displaystyle a_n={1\over n}+{1\over n+1}+\ldots+{1\over 2n}\)

    convergent?

    Difficulty: 7.


        Recursively Defined Sequences (semester 1, week 5; 21 problems)
        The Number $e$ (semester 1, week 5; 13 problems)
        Bolzano–Weierstrass Theorem and Cauchy Criterion (semester 1, week 6; 5 problems)
        Infinite Sums: Introduction (semester 1, week 6; 21 problems)
    Cardinalities of Sets (semester 1, week 7; 0 problems)
        Countable and not countable sets (semester 1, week 7; 6 problems)
        Not countable Sets (semester 1, week 7; 0 problems)
    Limit and Continuity of Real Functions (semester 1, weeks 7-10; 0 problems)
        Global Properties of Real Functions (semester 1, week 7; 20 problems)
        Continuity and Limits of Functions (semester 1, week 8; 33 problems)
        Calculating Limits of Functions (semester 1, week 8; 29 problems)
        Continuity and Convergent Sequences (semester 1, week 9; 0 problems)
        Continuous Functions on a Closed Bounded Interval (semester 1, week 9; 9 problems)
        Uniformly Continuous Functions (semester 1, week 10; 5 problems)
        Monotonity and Continuity (semester 1, week 10; 2 problems)
        Convexity and Continuity (semester 1, week 10; 7 problems)
    Elementary functions (semester 1, weeks 11-12; 0 problems)
        Arclength of the Graph of the Function (semester 1, week 11; 0 problems)
        Exponential, Logarithm, and Power Functions (semester 1, week 11; 17 problems)
        Inequalities (semester 1, week 12; 1 problems)
        Trigonometric Functions and their Inverses (semester 1, week 12; 3 problems)
    Differential Calculus and its Applications (semester 2, weeks 0-3; 0 problems)
        The Notion of Differentiation (semester 2, week 0; 47 problems)
        Tangents (semester 2, week 1; 11 problems)
        Higher Order Derivatives (semester 2, week 1; 13 problems)
        Local Properties and the Derivative (semester 2, week 1; 4 problems)
        Mean Value Theorems (semester 2, week 1; 3 problems)
        Number of Roots (semester 2, week 1; 5 problems)
        Exercises for Extremal Values (semester 2, week 2; 2 problems)
            Inequalities, Estimates (semester 2, week 2; 14 problems)
        The L'Hospital Rule (semester 2, week 2; 14 problems)
        Polynomial Approximation, Taylor Polynomial (semester 2, week 3; 20 problems)
        Convexity (semester 2, week 3; 5 problems)
        Analysis of Differentiable Functions (semester 2, week 3; 6 problems)
    Riemann Integral (semester 2, weeks 4-11; 0 problems)
        Definite Integral (semester 2, week 4; 11 problems)
        Indefinite Integral (semester 2, weeks 5-6; 13 problems)
        Properties of the Derivative (semester 2, week 6; 2 problems)
        Newton-Leibniz formula (semester 2, week 6; 2 problems)
        Integral Calculus (semester 2, week 7; 5 problems)
        Applications of the Integral Calculus (semester 2, week 8; 4 problems)
            Calculating the Area and the Volume (semester 2, week 8; 0 problems)
            Calculating the Arclength (semester 2, week 8; 3 problems)
            Surface Area of Surfaces of Revolution (semester 2, week 8; 0 problems)
        Integral and Inequalities (semester 2, week 8; 5 problems)
        Improper Integral (semester 2, week 9; 9 problems)
        Liouville Theorem (semester 2, week 10; 0 problems)
        Functions of Bounded Variation (semester 2, week 11; 2 problems)
        Riemann-Stieltjes integral (semester 2, week 11; 2 problems)
    Infinite Series (semester 2, weeks 12-13; 38 problems)
    Sequences and Series of Functions (semester 3, weeks 1-2; 0 problems)
        Convergence of Dequences of Functions (semester 3, week 1; 16 problems)
        Convergence of Series of Functions (semester 3, week 1; 17 problems)
        Taylor and Power Series (semester 3, week 2; 12 problems)
    Differentiability in Higher Dimensions (semester 3, weeks 3-6; 0 problems)
        Topology of the $n$-dimensional Space (semester 3, week 3; 29 problems)
        Real Valued Functions of Several Variables (semester 3, weeks 3-4; 0 problems)
            Limits and Continuity in $R^n$ (semester 3, week 3; 16 problems)
            Differentiation in $R^n$ (semester 3, week 4; 62 problems)
        Vector Valued Functions of Several Variables (semester 3, weeks 5-6; 0 problems)
            Limit and Continuity (semester 3, week 5; 3 problems)
            Differentiation (semester 3, week 5; 11 problems)
            Implicite functions (semester 3, week 6; 0 problems)
    Jordan Measure and Riemann Integral in Higher Dimensions (semester 3, weeks 7-9; 60 problems)
    Integral Theorems of Vector Calculus (semester 3, week 10 -- semester 4, week 1; 0 problems)
        The Line Integral (semester 3, week 10; 11 problems)
        Newton-Leibniz Formula (semester 3, week 11; 6 problems)
        Existence of the Primitive Function (semester 3, week 12; 13 problems)
        Integral Theorems in 2D (semester 4, week 1; 2 problems)
        Integral Theorems in 3D (semester 4, week 1; 12 problems)
    Measure Theory (semester 4, weeks 3-99; 0 problems)
        Set Algebras (semester 4, week 3; 9 problems)
        Measures and Outer Measures (semester 4, week 4; 8 problems)
        Measurable Functions. Integral (semester 4, week 5; 10 problems)
        Integrating Sequences and Series of Functions (semester 4, weeks 7-8; 11 problems)
        Fubini Theorem (semester 4, week 9; 1 problems)
        Differentiation (semester 4, weeks 11-12; 7 problems)
Complex Analysis (semester 5)
    Complex differentiability (semester 5, week 0; 0 problems)
        Complex numbers (semester 5, week 0; 21 problems)
            The Riemann sphere (semester 5, week 0; 1 problems)
    Regular functions (semester 5, weeks 1-2; 0 problems)
        Complex differentiability (semester 5, week 1; 7 problems)
        The Cauchy-Riemann equations (semester 5, week 1; 3 problems)
        Power series (semester 5, weeks 1-2; 0 problems)
            Domain of convergence (semester 5, week 2; 9 problems)
            Regularity of power series (semester 5, week 2; 2 problems)
            Taylor series (semester 5, week 2; 1 problems)
        Elementary functions (semester 5, week 2; 0 problems)
            The complex exponential and trigonometric functions (semester 5, week 2; 8 problems)
            Complex logarithm (semester 5, week 2; 12 problems)
    Complex Line Integral and Applications (semester 5, weeks 3-5; 0 problems)
        The complex line integral (semester 5, week 3; 9 problems)
        Cauchy's theorem (semester 5, week 3; 6 problems)
        The Cauchy formula (semester 5, week 4; 12 problems)
        Power and Laurent series expansions (semester 5, week 5; 0 problems)
            Power series expansion (semester 5, week 5; 2 problems)
            Liouville's Theorem (semester 5, week 5; 7 problems)
        Local properties of holomorphic functions (semester 5, week 5; 0 problems)
        Consequences of analyticity (semester 5, week 5; 8 problems)
            The maximum principle (semester 5, week 5; 4 problems)
        Laurent series (semester 5, week 5; 9 problems)
    Isolated singularities (semester 5, weeks 5-8; 0 problems)
        Singularities (semester 5, week 5; 4 problems)
        Cauchy's theorem on residues (semester 5, weeks 7-8; 15 problems)
            Residue calculus (semester 5, week 7; 6 problems)
            Applications (semester 5, week 7; 0 problems)
                Evaluation of series (semester 5, week 7; 7 problems)
                Evaluation of integrals (semester 5, week 7; 26 problems)
            The argument principle and Rouche's theorem (semester 5, week 8; 7 problems)
    Conformal maps (semester 5, weeks 9-10; 0 problems)
        Fractional linear transformations (semester 5, week 9; 20 problems)
        Riemann mapping theorem (semester 5, week 9; 11 problems)
        Schwarz lemma (semester 5, week 9; 12 problems)
        Caratheodory's theorem (semester 5, week 9; 2 problems)
        Schwarz reflection principle (semester 5, week 10; 2 problems)
    Harmonic functions (semester 5, weeks 11-12; 8 problems)
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