MathDB

13

Part of 2016 ASDAN Math Tournament

Problems(2)

2016 Guts #13

Source:

8/14/2022
Suppose {an}n=1∞\{a_n\}_{n=1}^\infty is a sequence. The partial sums {sn}n=1∞\{s_n\}_{n=1}^\infty are defined by sn=∑i=1nai.s_n=\sum_{i=1}^na_i. The Cesàro sums are then defined as {An}n=1∞\{A_n\}_{n=1}^\infty, where An=1n⋅∑i=1nsi.A_n=\frac{1}{n}\cdot\sum_{i=1}^ns_i. Let an=(−1)n+1a_n=(-1)^{n+1}. What is the limit of the Cesàro sums of {an}n=1∞\{a_n\}_{n=1}^\infty as nn goes to infinity?
2016Guts Round
2016 Team #13

Source:

8/17/2022
Ash writes the positive integers from 11 to 20162016 inclusive as a single positive integer n=1234567891011…2016n=1234567891011\dots2016. What is the result obtained by successively adding and subtracting the digits of nn? (In other words, compute 1−2+3−4+5−6+7−8+9−1+0−1+…1-2+3-4+5-6+7-8+9-1+0-1+\dots.)
2016team test