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) \(\displaystyle \frac{5-2n^2}{4+n}\), 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) \(\displaystyle \sqrt[n]{\frac{1}{n}-\frac{2}{n^2}}\), 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) \(\displaystyle \sqrt{n^2 +2} +\sqrt{n^2 -2} -2n;\) 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: 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. |
Supported by the Higher Education Restructuring Fund allocated to ELTE by the Hungarian Government |