\(\displaystyle \sum_{i=1}^{\infty}\left({1\over 2^i}+{1\over 3^i}\right)=?\)

Difficulty: 2.

Suppose that \(\displaystyle \sum a_n\) is convergent. Show that \(\displaystyle \lim (a_{n+1}+a_{n+1}+\ldots+a_{n^2})=0\).

Difficulty: 2.

Convergent or divergent?

\(\displaystyle \sum {n^{100} \over 1.001^n}\)

Difficulty: 3.

Convergent or divergent?

\(\displaystyle \sum {1\over \sqrt{(2i-1)(2i+1)}}\)

Difficulty: 3.

Convergent or divergent?

\(\displaystyle a)\ \sum_{n=1}^{\infty} {1\over n(n+1)(n+2)}\qquad\qquad b)\ \sum_{n=1}^{\infty} {n^2\over (2+{1\over n})^n}\)

Difficulty: 3.

\(\displaystyle \sum_{n=1}^{\infty} {1\over n(n+1)}=?\)

Difficulty: 4.

Find a sequence \(\displaystyle a_n\) such that \(\displaystyle \sum a_n\) is convergent, and \(\displaystyle a_{n+1}/a_n\) is not bounded.

Difficulty: 4. Solution is available for this problem.

Convergent or divergent?

\(\displaystyle {1000 \over 1} +{1000\cdot 1001 \over 1\cdot 3}+{1000\cdot 1001\cdot 1002 \over 1\cdot 3\cdot 5}+\ldots\)

Difficulty: 4.

Convergent or divergent?

\(\displaystyle \sum {7^n\over \sqrt{n!}}\)

Difficulty: 4.

For which \(\displaystyle x\) is the sum

\(\displaystyle \sum{x^n\over a^n+b^n}\)

convergent?

Difficulty: 4.

For which \(\displaystyle x\) the sum

\(\displaystyle \sum \log\left({k+1\over k}\right)x^k\)

is convergent?

Difficulty: 4.

\(\displaystyle \sum_{n=1}^{\infty} {1\over n^2-3n+{1\over 2}}=?\)

Difficulty: 5.

Prove that

\(\displaystyle \sum_{n=1}^{\infty} {1\over n^2}<2.\)

Difficulty: 5. Solution is available for this problem.

Convergent or divergent?

\(\displaystyle \sum_{n=1}^{\infty} (\sqrt[n]{e}-1)\)

Difficulty: 5.

Show that if \(\displaystyle |a_{n+1}-a_n|<{1\over n^2}\) then \(\displaystyle (a_n)\) is convergent.

Difficulty: 5.

For which \(\displaystyle x\) and \(\displaystyle p\) is the sum

\(\displaystyle \sum{x^n\over n^p}\)

convergent?

Difficulty: 5.

(a) Prove that if \(\displaystyle \liminf{\log {1\over a_k} \over \log k}>1\), then \(\displaystyle \sum a_k\) is convergent.

(b) Prove that if \(\displaystyle \limsup{\log {1\over a_k} \over \log k}<1\), then \(\displaystyle \sum a_k\) is divergent.

(c) Construct a sequence \(\displaystyle a_n\) such that \(\displaystyle {\log {1\over a_k} \over \log k}\to1\), and \(\displaystyle \sum a_k\) convergent.

(d) Construct a sequence \(\displaystyle a_n\) such that \(\displaystyle {\log {1\over a_k} \over \log k}\to1\), and \(\displaystyle \sum a_k\) divergent.

Difficulty: 5.

Convergent or divergent?

\(\displaystyle \sum {(2k)! \over 4^k(k!)^2}\)

Difficulty: 6.

Convergent or divergent?

\(\displaystyle \sum {(2k)! \over 4^k(k!)^2} {1\over 2k+1}\)

Difficulty: 6.

For which \(\displaystyle z\in \C\) is the following sum convergent?

\(\displaystyle \sum z^n \qquad \sum {z^n\over n} \qquad \sum {z^n\over n^2}\)

Difficulty: 7.

\(\displaystyle h_n:=\sum_{i=1}^{n} {1\over i}\). Prove that

\(\displaystyle {1\over h_1^2}+{1\over 2h_2^2}+\ldots+{1\over nh_n^2}<2.\)

Difficulty: 7.