Prove or disprove the following statement: There exists a function whose Maclaurin series converges at only one point.
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Category Archives: problem
2017-20 Convergence of a series
Determine whether or not the following infinite series converges. \[ \sum_{n=0}^{\infty} \frac{ 1 }{2^{2n}} \binom{2n}{n}.\]
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2017-19 Identity
For an integer \( p \), define
\[
f_p(n) = \sum_{k=1}^n k^p.
\]
Prove that
\[
\frac{1}{2} \sum_{n=1}^{\infty} \frac{f_{-1}(n)}{f_3(n)} + 2\sum_{n=1}^{\infty} \frac{f_{-1}(n)}{f_1(n)} = \sum_{n=1}^{\infty} \frac{(f_{-1}(n))^2}{f_1(n)}.
\]
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2017-18 Limit
Suppose that \(f\) is differentiable and \[ \lim_{x\to\infty} (f(x)+f'(x))=2.\] What is \( \lim_{x\to\infty} f(x)\)?
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2017-17 An infimum
For an integer \( n \geq 3 \), evaluate
\[
\inf \left\{ \sum_{i=1}^n \frac{x_i^2}{(1-x_i)^2} \right\},
\]
where the infimum is taken over all \( n \)-tuple of real numbers \( x_1, x_2, \dots, x_n \neq 1 \) satisfying that \( x_1 x_2 \dots x_n = 1 \).
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2017-16 Finding a rectangle
Is it possible to color all lattice points (\(\mathbb Z\times \mathbb Z\)) in the plane into two colors such that if four distinct points \( (a,b), (a+c,b), (a,b+d), (a+c,b+d)\) have the same color, then \( d/c\notin \{1,2,3,4,6\}\)?
(The next POW problem will be posted on October 20. Happy Chuseok and good luck with your midterm exams.)
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2017-15 Infinite product
For \( x \in (1, 2) \), prove that there exists a unique sequence of positive integers \( \{ x_i \} \) such that \( x_{i+1} \geq x_i^2 \) and
\[
x = \prod_{i=1}^{\infty} (1 + \frac{1}{x_i}).
\]
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2017-14 Polynomials of degree at most n
Let \(f(x)\in \mathbb R[x]\) be a polynomial of degree at most \(n\) such that \[ x^2+f(x)^2\le 1\] for all \( -1\le x\le 1 \). Prove that \( \lvert f'(x)\rvert \le 2(n-1)\) for all \( -1\le x\le 1\).
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2017-13 Infinite series with recurrence relation
Let \(a_0 = a_1 =1\) and \(a_n = n a_{n-1} + (n-1) a_{n-2}\) for \(n \geq 2\). Find the value of
\[
\sum_{n=0}^{\infty} (-1)^n \frac{n!}{a_n a_{n+1}}.
\]
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2017-12 Invertible matrices
Let \(A\) and \(B\) be \(n\times n\) matrices. Prove that if \(n\) is odd and both \(A+A^T\) and \(B+B^T\) are invertible, then \(AB\neq 0\).
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