[College Calc 01-02. Limits of Sequences] #13

Calculus Level 4

Which of the followings are true for any bounded sequences { a n } , { b n } \{a_n\},~\{b_n\} of real numbers?

A . lim sup ⁡ n → ∞ a n = lim sup ⁡ n → ∞ a 3 n . B . lim sup ⁡ n → ∞ ( a n + b n ) = lim sup ⁡ n → ∞ a n + lim sup ⁡ n → ∞ b n . C . lim sup ⁡ n → ∞ ( a n + 1 − a n ) = 0. \begin{aligned} {\rm A.} &&& \limsup_{n\to\infty} a_n = \limsup_{n\to\infty} a_{3n}. \\ {\rm B.} &&& \limsup_{n\to\infty} (a_n+b_n)=\limsup_{n\to\infty} a_n+\limsup_{n\to\infty} b_n. \\ {\rm C.} &&& \limsup_{n\to\infty}(a_{n+1}-a_n)=0. \end{aligned}

This is a part of the College Calc problem set. You can find more problems here .

A \rm A only C \rm C only B , C \rm B,~C only None of them All of them B \rm B only A , C \rm A,~C only A , B \rm A,~B only

Boi (보이) by Boi (보이) , 19, South Korea

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

1 solution

Boi (보이)
Dec 24, 2020

A. False

Consider the sequence { a n } : = 3 , 2 , 1 , 3 , 2 , 1 , 3 , 2 , 1 , ⋯ . \{a_n\}:=3,2,1,3,2,1,3,2,1,\cdots. We have

lim sup ⁡ n → ∞ a n = 3 ; lim sup ⁡ n → ∞ a 3 n = 1. \limsup_{n\to\infty}a_n = 3;\quad \limsup_{n\to\infty}a_{3n}=1.

B. False

Consider the sequences { a n } : = 1 , 2 , 1 , 2 , 1 , 2 , ⋯ \{a_n\}:=1,2,1,2,1,2,\cdots and { b n } : = 2 , 1 , 2 , 1 , 2 , 1 , ⋯ . \{b_n\}:=2,1,2,1,2,1,\cdots. We have

lim sup ⁡ n → ∞ a n = lim sup ⁡ n → ∞ b n = 2 ; lim sup ⁡ n → ∞ ( a n + b n ) = 3. \limsup_{n\to\infty}a_n=\limsup_{n\to\infty}b_n=2;\quad \limsup_{n\to\infty}(a_n+b_n)=3.

C. False

Consider the sequence { a n } : = 0 , 1 , 0 , 1 , 0 , 1 , ⋯ . \{a_n\}:=0,1,0,1,0,1,\cdots. We have that { a n + 1 − a n } = 1 , − 1 , 1 , − 1 , 1 , − 1 , ⋯ , \{a_{n+1}-a_n\}=1,-1,1,-1,1,-1,\cdots, such that

lim sup ⁡ n → ∞ ( a n + 1 − a n ) = 1. \limsup_{n\to\infty}(a_{n+1}-a_n)=1.

2 Helpful 2 Interesting 2 Brilliant 0 Confused

Oh, I confused bounded with convergent and proved all of them true. Seeing the answer None of them, I was like wait, wtf??!!

Veselin Dimov - 5 months, 2 weeks ago

Log in to reply

Haha well yeah the problem's meant to be confusing

Boi (보이) - 5 months, 2 weeks ago

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...