Show that

\(\displaystyle {1\over n+1}<\log(n+1)-\log(n)<{1\over n}.\)

Difficulty: 1.

\(\displaystyle \frac1{1\cdot2}+ \frac1{2\cdot3}+ \frac1{3\cdot4}+ \frac1{4\cdot5}+\ldots=? \)

Difficulty: 2.

Prove that if \(\displaystyle \sum a_n\) and \(\displaystyle \sum b_n\) are absolutely convergent, then the following series are also absolutely convergent:

\(\displaystyle \sum(a_n+b_n) \qquad \sum\max(a_n, b_n) \qquad \sum\sqrt{a_n^2+b_n^2} \)

Difficulty: 2.

Prove

\(\displaystyle {1\over n}\le 1+{1\over 2}+{1\over 3}+\ldots+{1\over n}-\log n<1.\)

Difficulty: 3.

Determine whether the following series are convergent or divergent. In case of convergence determine whether convergence is absolute or conditional.

\(\displaystyle \sum_{n=1}^\infty \frac{1}{10n+\sqrt{n}+1} \qquad \sum_{n=1}^\infty \frac{1}{n^2} \qquad \sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} \qquad \sum_{n=1}^\infty \frac{(-1)^{[n/2]}}{\log(n+1)} \qquad \sum_{n=1}^\infty \frac{1}{n!} \)

Difficulty: 3.

Determine whether the following series are convergent or divergent.

\(\displaystyle \sum e^{-n^2} \qquad \sum\frac{n^{10}}{3^n-2^n} \qquad \sum\frac1{\sqrt{n(n+1)}} \qquad \sum n^2 e^{-\sqrt{n}} \qquad \sum \left(n^{1/n^2}-1\right) \qquad \sum \frac{\root{n}\of{n}-1}{\log^2n} \)

Difficulty: 3.

Determine whether the following series are convergent or divergent. In case of convergence, determine whether the convergence is absolute or conditional.

\(\displaystyle \sum_{n=1}^\infty \frac{(-1)^n}{n\log(n+1)} \qquad \sum_{n=1}^\infty \frac{(n!)^2}{2^{n^2}} \qquad \sum_{n=1}^\infty \frac{(-1)^n(n!)^2}{2^{n^2}} \qquad \sum_{n=1}^\infty \frac{1}{\binom{2n}{n}} \)

Difficulty: 3.

\(\displaystyle 1-{1\over 2}+{1\over 3}-{1\over 4}+{1\over 5}-{1\over 6}+{1\over 7}-{1\over 8}+\ldots=?\)

Difficulty: 4.

\(\displaystyle 1+{1\over 3}-{1\over 2}+{1\over 5}+{1\over 7}-{1\over 4}+ {1\over 9}+{1\over 11}-{1\over 6}+\ldots=?\)

Difficulty: 4.

\(\displaystyle 1-{1\over 2}-{1\over 4}+{1\over 3}-{1\over 6}-{1\over 8} +{1\over 5}-{1\over 10}-{1\over 12}+\ldots=?\)

Difficulty: 4.

\(\displaystyle 1+{1\over 2}-{1\over 3}+{1\over 4}+{1\over 5}-{1\over 6} +{1\over 7} +{1\over 8}-{1\over 9}+\ldots=?\)

Difficulty: 4.

\(\displaystyle \sum\limits_{n=0}^\infty (n+1)q^n=? \)

Difficulty: 4.

True or False?

(a) If \(\displaystyle a_n\to0\), then \(\displaystyle \sum\limits_{n=1}^\infty a_n\) is convergent.

(b) If \(\displaystyle a_n\to0\) and the partial sums \(\displaystyle \sum\limits_{n=1}^\infty a_n\) are bounded, then \(\displaystyle \sum\limits_{n=1}^\infty a_n\) is convergent.

(c) If \(\displaystyle \sum\limits_{n=1}^\infty a_n\) is convergent, then \(\displaystyle a_n\to0\).

Difficulty: 4.

Show that if \(\displaystyle |a_n|<\frac1{n^2}\) for all positive integer \(\displaystyle n\), then \(\displaystyle \sum a_n\) satisfies the Cauchy-criterion.

Difficulty: 4.

\(\displaystyle \frac1{1\cdot2\cdot3}+ \frac1{2\cdot3\cdot4}+ \frac1{3\cdot4\cdot5}+ \frac1{4\cdot5\cdot6}+ \ldots=? \)

Difficulty: 4.

Assume that \(\displaystyle a_n\le b_n\le c_n\) for all positive integer \(\displaystyle n\). Show that if \(\displaystyle \sum\limits_{n=1}^\infty a_n\) and \(\displaystyle \sum\limits_{n=1}^\infty c_n\) are convergent, then \(\displaystyle \sum\limits_{n=1}^\infty b_n\) is also convergent.

Difficulty: 4.

Determine whether the following series are convergent or divergent.

\(\displaystyle \sum \left(1-\frac1n\right)^n \qquad \sum \left(1-\frac1n\right)^{n^2} \qquad \sum \left(\frac{n-1}{n+1}\right)^{\frac{n}2\log n+n\log\log n} \qquad \sum \frac{n^{n+\frac1n}}{\left(n+\frac1n\right)^n} \)

Difficulty: 4.

For which \(\displaystyle c \in \R\) is the series

\(\displaystyle \sum_{n=10}^\infty\dfrac1{n\cdot\log n\cdot(\log\log n)^c} \)

convergent?

Difficulty: 4.

Prove that

\(\displaystyle a_n:=1+{1\over 2}+{1\over 3}+\ldots+{1\over n}-\log n\)

is convergent.

Difficulty: 5.

Let \(\displaystyle u_n:=\int_0^{1/n} {\sqrt{x} \over 1+x^2}\dx\). Is the series \(\displaystyle \sum\limits_1^{\infty} u_n\) convergent?

Difficulty: 5.

\(\displaystyle \sum\limits_{n=0}^\infty n^2q^n=? \)

Difficulty: 5.

Assume that \(\displaystyle a_n>0\), \(\displaystyle b_n>0\) for all \(\displaystyle n\) and that \(\displaystyle a_n/b_n\to1\). Prove that \(\displaystyle \sum a_n\) is convergent if and only if \(\displaystyle \sum b_n\) is convergent. Give an example when this fails if the assumption \(\displaystyle a_n>0,b_n>0\) is removed.

Difficulty: 5.

What are the root test, quotient test, Dirichlet-test, and Abel-test for improper integrals?

Difficulty: 5.

(a) Show that if \(\displaystyle \displaystyle \Olim \left( \big|a_n\big|^{\frac1{\log n}} \right) < \frac1e\), then \(\displaystyle \displaystyle\sum_{n=1}^\infty a_n\) is absolutely convergent.

(b) Show that if \(\displaystyle a_n\ge0\) and \(\displaystyle \displaystyle \Ulim \left( \big|a_n\big|^{\frac1{\log n}} \right) > \frac1e \), then \(\displaystyle \displaystyle\sum_{n=1}^\infty a_n\) is divergent.

(c) Can any conclusions be made about the convergence of \(\displaystyle \displaystyle\sum_{n=1}^\infty a_n\) if \(\displaystyle a_n>0\) and \(\displaystyle \displaystyle \lim \left( \big|a_n\big|^{\frac1{\log n}} \right) =\frac1e \)?

Difficulty: 5.

Prove the Condensation lemma: Let \(\displaystyle a_1\geq a_2\geq\cdots \geq a_n\geq\cdots\geq 0\). Then

\(\displaystyle \sum_{n=1}^\infty a_n \qquad \text{convergent} \iff \sum_{k=1}^\infty 2^ka_{2^k} \qquad \text{convergent}. \)

Difficulty: 5. Solution is available for this problem.

Using Dirichlet's criterion show that \(\displaystyle \sum\limits_{n=1}^\infty \dfrac{\sin(na)}{n}\) converges for all \(\displaystyle a\in \R\).

Difficulty: 5.

True or false?

(1) If \(\displaystyle \sum\limits_{n=1}^\infty a_n\) is convergent, then \(\displaystyle \sum\limits_{n=1}^\infty (\root{n}\of{2}\cdot a_n)\) is also convergent.

(2) If \(\displaystyle \sum\limits_{n=1}^\infty a_n\) is divergent, then \(\displaystyle \sum\limits_{n=1}^\infty (\root{n}\of{2}\cdot a_n)\) is also divergent.

(3) If \(\displaystyle \sum\limits_{n=1}^\infty a_n\)is convergent, then \(\displaystyle \sum\limits_{n=1}^\infty \dfrac{a_n}{n}\) is also convergent.

(4) If \(\displaystyle \sum\limits_{n=1}^\infty a_n\) is divergent, then \(\displaystyle \sum\limits_{n=1}^\infty \dfrac{a_n}{n}\) is also divergent.

Difficulty: 5.

Give examples of an absolutely convergent series \(\displaystyle \sum\limits_{n=0}^\infty a_n\) and conditionally convergent series \(\displaystyle \sum\limits_{n=0}^\infty b_n\) for which their Cauchy product is conditionally convergent.

Difficulty: 5.

Let \(\displaystyle \sum\limits_{n=1}^\infty a_n\) has positive terms.

(a) Prove that if \(\displaystyle \displaystyle \liminf~ n\left(\frac{a_n}{a_{n+1}}-1\right)>1\), then the series is convergent.

(b) Prove that if \(\displaystyle \displaystyle n\left(\frac{a_n}{a_{n+1}}-1\right)\le1\) for \(\displaystyle n\) large enough, then the series is divergent.

(Raabe-criterion)

Difficulty: 5.

Let \(\displaystyle \sum a_{\varphi(n)}\) be a rearrangment of the conditionally convergent series \(\displaystyle \sum a_n\). What can be the set of limit points of the set of the partial sums \(\displaystyle \sum\limits_{k=1}^na_{\varphi(k)}\)?

Difficulty: 6.

Convergent or divergent?

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

Difficulty: 6.

Let \(\displaystyle \varepsilon >0\). Convergent or divergent?

\(\displaystyle \sum_{n=2}^\infty {1\over n (\log n)^{1+\varepsilon}}\)

Difficulty: 6.

Let \(\displaystyle a_1,a_2,\ldots\) be a sequence of positive reals such that

\(\displaystyle \exists c>0 \quad \forall x>2 \quad \big|\{k:~a_k<x\}\big|>c\frac{x}{\log x}. \)

(Primes for example staisfiy this.) Show that \(\displaystyle \sum\frac1{a_k}=\infty\).

Difficulty: 7.

Let \(\displaystyle \sum\limits_{n=1}^n a_n\) be a divergent series with positive terms. Prove that there is a sequence \(\displaystyle c_n\) of positive numbers, such that \(\displaystyle c_n \to 0\) as \(\displaystyle n \to \infty\) and \(\displaystyle \sum\limits_{n=1}^n (c_n\cdot a_n)\) still diverges.

Difficulty: 8.

Let \(\displaystyle \sum\limits_{n=1}^n a_n\) be a convergent series of positive terms. Prove that there is a sequence \(\displaystyle (c_n)\) such that \(\displaystyle c_n \to \infty\) as \(\displaystyle n \to \infty\) and for which \(\displaystyle \sum\limits_{n=1}^n (c_n\cdot a_n)\) is still convergent.

Difficulty: 8.

For \(\displaystyle s>1\) let \(\displaystyle \zeta(s)=\sum\limits_{n=1}^\infty\frac1{n^s}\), \(\displaystyle (p_1,p_2,p_3,\ldots)=(2,3,5,\ldots)\) be the sequence of primes in increasing order.

(a) Prove that \(\displaystyle \displaystyle \lim_{N\to\infty} \prod_{n=1}^N \dfrac1{1-\frac1{p_n^s}} = \zeta(s) \).

(b) Prove that \(\displaystyle \sum\limits_{n=1}^\infty\frac1{p_n}=\infty\).

(c) What is the order of magnitude of \(\displaystyle \displaystyle \sum\limits_{n=1}^\infty \frac1{p_n^s}\) as \(\displaystyle s\to1+0\)?

Difficulty: 8.

For all \(\displaystyle k \in \N\) let \(\displaystyle \sum\limits_{n=1}^\infty a^{(k)}_n\) be a divergent series of positive terms. Prove that there is a sequence \(\displaystyle (c_n)\) of positive real numbers such that the series \(\displaystyle \sum\limits_{n=1}^\infty (c_n\cdot a^{(k)}_n)\) are all divergent.

Difficulty: 9.

For a sequence \(\displaystyle A=(a_0,a_1,a_2,\ldots)\) of reals let

\(\displaystyle SA=(a_0,a_0+a_1,a_0+a_1+a_2,\ldots) \)

be the sequence of its partial sums \(\displaystyle a_0+a_1+a_2+\ldots\). Can one find a non-zero sequence \(\displaystyle A\) for which the sequences \(\displaystyle A\), \(\displaystyle SA\), \(\displaystyle SSA\), \(\displaystyle SSSA\), ...are all convergent?

Miklós Schweitzer memorial competition, 2007

Difficulty: 10.