Boundedness of a Sequence

Question : How do we prove that the sequence (1+(1/n))^n is bounded?

My thoughts on this question : We know that, to prove a sequence is bounded, we should prove that it is both bounded above and bounded below.

We can see that all the terms in the sequence is greater than 0. Which means the sequence is bounded below.
But gow do we prove that it is bounded above. Does taking the limit of the general term(say nth term) as n tends to infinity always gives us the highest possible term of the sequence, which will give us the opportunity to find an upper bound of the sequence?

Thanks!

1. 👍
2. 👎
3. 👁
1. Hmmm. By the Binomial Theorem,
(1 + 1/n)^n
= 1 + n*1/n + n(n-1)/2!n^2 + n(n-1)(n-2)/3!n^2 + ...
< 1 + 1 + 1/2! + 1/3! + ...
which you can show converges

I expect google can turn up various more direct proofs.

1. 👍
2. 👎
👤
oobleck
2. I Googled this and find that proof. But can you please let me know whether we can take the highest value of a monotonically increasing sequence by taking the limit of its general term(say nth term), when n goes to infinity?
Can we consider this value(if finite), as an upper bound of this sequence?

1. 👍
2. 👎

Similar Questions

1. Math

this is a difficult question for me please help! thankyou A sequence is defined recursively by an + 1 = 3an − n, a1 = 2. Find the first six terms of the sequence. a1 = a2 = a3 = a4 = a5 = a6 =

2. mathematics

The following sequence of numbers forms a quadratic sequence: –3, –2, –3, –6, –11, …... The first differences of the sequence also form a sequence. Determine an expression for general term of the first differences

Question 1 Write the first four terms of the sequence whose general term is given. an = 3n - 1 Answer 2, 3, 4, 5 2, 5, 8, 11 -2, -5, -8, -11 4, 7, 10, 13 3 points Question 2 Write the first four terms of the sequence whose general

4. math

term given by x+2,3x-1,and 4x-1 form an arithmetic sequence. find the value of x. i didn't get it how to start this question and how it forms arithmetic sequence?

1. real analysis

definiton: A set S subsect of real number, sequence {Xn} is called bounded if there is a number M such that |X|≤ M for all x ∈ S prove the sequence (-8+(n/(n+1))) is bounded. my anwser proof: Let Xn = (-8+(n/(n+1))) there

2. Math

Alpha writes the infinite arithmetic sequence 10, 8, 6, 4, 2, 0... Beta writes the infinite geometric sequence 9, 6, 4, 8/3, 16/9,... Gamma makes a sequence whose n^th term is the product of the n^th term of Alpha's sequence and

3. Sequence and series

If {gn} is a geometric sequenc with gn > 0 for all n is an element of natural number,then prove that {ln gn} is an arthnetic sequence

4. Math

an = 3n - 1 Answer 2, 3, 4, 5 2, 5, 8, 11 -2, -5, -8, -11 4, 7, 10, 13 3 points Question 2 Write the first four terms of the sequence whose general term is given. an = 2(2n - 3) Answer -6, -2, 2, 6 -1, 1, 3, 5 -2, -4, -6, -8 -2,

1. Math

Question 1 Write the first four terms of the sequence whose general term is given. an = 3n - 1 Answer 2, 3, 4, 5 2, 5, 8, 11 -2, -5, -8, -11 4, 7, 10, 13 3 points Question 2 Write the first four terms of the sequence whose general

2. pre cal 11

hi so i am stuck at this question that is asking about arithmetic sequence . Five fence posts are equally spaced between two corner posts that are 42 meter apart. how far apart are the five fence posts. i know the answer is 7m but

3. Modern History

Hi Guys! Okay sweet, here goes. I'm supposed to be keeping a research journal of my research. In it I like examine my research and process. I've already done my Hypothesis. Under the topic of INITIAL THOUGHTS AND IDEAS I outlined

4. Quick math help

Find the tenth term of the sequence: -6,1,8... Is it 57? For the sequence:2,4,8,16 the value of s4 is____. Is it 8? Find the 7th term of the sequence: 1,2,4,... Is it 64? Which term of this sequence is 275? 5,10,15,... Is it 1357?