# Solution: 2017-21 Maclaurin series

Prove or disprove the following statement: There exists a function whose Maclaurin series converges at only one point.

The best solution was submitted by Kook, Yun Bum (국윤범, 수리과학과 2015학번). Congratulations!

Here is his solution of problem 2017-21.

Alternative solutions were submitted by 민찬홍 (중앙대학교사범대학부속고등학교 3학년, +3), 채지석 (수리과학과 2016학번, +3), 최대범 (수리과학과 2016학번, +3), 하석민 (2017학번, +3), Huy Tung Nguyen (수리과학과 2016학번, +3). Four incorrect solutions were submitted, mostly due to misunderstanding.

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# 2017-22 Debugging

Let $$p$$, $$q$$, $$r$$ be positive integers such that $$p,q\ge r$$. Ada and Betty independently read all source codes of their programming project. Ada found $$p$$ bugs and Betty found $$q$$ bugs, including $$r$$ bugs that Ada found. What is the expected number of remaining bugs that neither Ada nor Betty found?

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# Solution: 2017-20 Convergence of a series

Determine whether or not the following infinite series converges. $\sum_{n=0}^{\infty} \frac{ 1 }{2^{2n}} \binom{2n}{n}.$

The best solution was submitted by Lee, Bonwoo (이본우, 2017학번). Congratulations!

Here is his solution of problem 2017-20.

Alternative solutions were submitted by 고성훈 (+3), 국윤범 (수리과학과 2015학번, +3), 길현준 (인천과학고등학교 2학년, +3), 김태균 (수리과학과 2016학번, +3), 민찬홍 (중앙대학교사범대학부속고등학교 3학년, +3), 유찬진 (수리과학과 2015학번, +3), 이원웅 (건국대 수학과 2014학번, +3), 이준협 (하나고등학교, +3), 채지석 (2016학번, +3), 최대범 (수리과학과 2016학번, +3), 하석민 (2017학번, +3), Huy Tung Nguyen (수리과학과 2016학번, +3), 이준성 (상문고등학교 1학년, +3), 정경훈 (서울대학교 컴퓨터공학과, +3), Mirali Ahmadili & Saba Dzmanashvili (2017학번, +3).

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# Solution: 2017-19 Identity

For an integer $$p$$, define
$f_p(n) = \sum_{k=1}^n k^p.$
Prove that
$\frac{1}{2} \sum_{n=1}^{\infty} \frac{f_{-1}(n)}{f_3(n)} + 2\sum_{n=1}^{\infty} \frac{f_{-1}(n)}{f_1(n)} = \sum_{n=1}^{\infty} \frac{(f_{-1}(n))^2}{f_1(n)}.$

The best solution was submitted by Kim, Taegyun (김태균, 수리과학과 2016학번). Congratulations!

Here is his solution of problem 2017-19.

Alternative solutions were submitted by 국윤범 (수리과학과 2015학번, +3), 길현준 (인천과학고등학교 2학년, +3), 민찬홍 (중앙대학교사범대학부속고등학교 3학년, +3), 유찬진 (수리과학과 2015학번, +3), 이본우 (2017학번, +3), 채지석 (2016학번, +3), 최대범 (수리과학과 2016학번, +3), Huy Tung Nguyen (수리과학과 2016학번, +3), 이재우 (함양고등학교 2학년, +2), 하석민 (2017학번, +2), Saba Dzmanashvili & Mirali Ahmadili  (2017학번, +2).

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# 2017-21 Maclaurin series

Prove or disprove the following statement: There exists a function whose Maclaurin series converges at only one point.

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# 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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# Solution: 2017-18 Limit

Suppose that $$f$$ is differentiable and $\lim_{x\to\infty} (f(x)+f'(x))=2.$  What is $$\lim_{x\to\infty} f(x)$$?

The best solution was submitted by You, Chanjin (유찬진, 수리과학과 2015학번). Congratulations!

Here is his solution of problem 2017-18.

Alternative solutions were submitted by 국윤범 (수리과학과 2015학번, +3), 김태균 (수리과학과 2016학번, +3), 민찬홍 (중앙대학교사범대학부속고등학교 3학년, +3), 이본우 (2017학번, +3), 장기정 (수리과학과 2014학번, +3, alternative solution), 채지석 (2016학번, +3), 최대범 (수리과학과 2016학번, +3), 하석민 (2017학번, +3), Huy Tung Nguyen (수리과학과 2016학번, +3), 윤준기 (전기및전자공학부 2014학번, +2). One incorrect solution was received.

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# Solution: 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 $$x_1 x_2 \dots x_n = 1$$.

The best solution was submitted by Choi, Daebeom (최대범, 수리과학과 2016학번). Congratulations!

Here is his solution of problem 2017-17.

Alternative solutions were submitted by 국윤범 (수리과학과 2015학번, +3), 김태균 (수리과학과 2016학번, +3), 장기정 (수리과학과 2014학번, +3), Huy Tung Nguyen (수리과학과 2016학번, +3), 김기택 (수리과학과 2015학번, +2), 유찬진 (수리과학과 2015학번, +2), 윤준기 (전기및전자공학부 2014학번, +2), 이본우 (2017학번, +2). One incorrect solution was received.

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For an integer $$p$$, define
$f_p(n) = \sum_{k=1}^n k^p.$
$\frac{1}{2} \sum_{n=1}^{\infty} \frac{f_{-1}(n)}{f_3(n)} + 2\sum_{n=1}^{\infty} \frac{f_{-1}(n)}{f_1(n)} = \sum_{n=1}^{\infty} \frac{(f_{-1}(n))^2}{f_1(n)}.$