Real Analysis Proof: r(n), t(n), e, and n

In summary, we have shown that t(n) > r(n) for all n and that the limit of (t(n) - r(n)) as n approaches infinity is 0. We have also shown that the sequence {tn} is decreasing and has a limit of e. Using Bernoulli's inequality, we were able to prove this. To estimate e to 3 decimal places, we can use n=10 to calculate upper and lower estimates. This problem appears to be related to calculus more than real analysis.
  • #1
teacher2love
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1. Let r(n) = (1+1/n)^n and t(n) = (1+1/n)^n+1. (Use r(n) converge to e).
Show that t(n) > r(n) for all n and that lim n->inf(t(n) - r(n)) = 0.
Show that {tn} is a decreasing sequence with limit e. {Hint: express {(1+1/n-1)/(1+1/n)}^n as (1+a)^n and apply Bernoulli's inequality). Use n=10 to calculate upper and lower estimates for e. How large should n be to estimate e to 3 decimal places?
 
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  • #2
teacher2love said:
1. Let r(n) = (1+1/n)^n and t(n) = (1+1/n)^n+1. (Use r(n) converge to e).
Show that t(n) > r(n) for all n and that lim n->inf(t(n) - r(n)) = 0.
Show that {tn} is a decreasing sequence with limit e. {Hint: express {(1+1/n-1)/(1+1/n)}^n as (1+a)^n and apply Bernoulli's inequality). Use n=10 to calculate upper and lower estimates for e. How large should n be to estimate e to 3 decimal places?

well about showing that t(n)>r(n) does not appear to be that difficult

[tex] (1+\frac{1}{n})^{n+1}=(1+\frac{1}{n})^{n}(1+\frac{1}{n})[/tex]
now since, [tex](1+\frac{1}{n})>1[/tex] we have that

[tex]t(n)= (1+\frac{1}{n})^{n+1}=(1+\frac{1}{n})^{n})(1+\frac{1}{n})>(1+\frac{1}{n})^{n}=r(n)[/tex]

now:

[tex]\lim_{n\rightarrow\infty}(t(n)-r(n))=\lim_{n\rightarrow\infty}((1+\frac{1}{n})^{n}(1+\frac{1}{n}))-\lim_{x\rightarrow\infty}(1+\frac{1}{n})^{n}=\lim_{n\rightarrow\infty}(1+\frac{1}{n})^{n}*\lim_{n\rightarrow\infty}(1+\frac{1}{n})-e=e*1-e=0[/tex]

Now we want to show that [tex]t_n-_1>t_n[/tex] this part is quite easy to show as well

[tex]\frac{t_n-_1}{t_n}=\frac{(1+\frac{1}{n-1})^{n}}{(1+\frac{1}{n})^{n+1}}=...=(\frac{n^{2}}{n^{2}-1})^{n}*\frac{n}{n+1}=(\frac{n^2-1+1}{n^{2}-1})^{n}\frac{n}{n+1}=(1+\frac{1}{n^{2}-1})^{n}\frac{n}{n+1}[/tex] now using bernuli inequality we get:

[tex]\frac{t_n-_1}{t_n}=\frac{(1+\frac{1}{n-1})^{n}}{(1+\frac{1}{n})^{n+1}}=...=(\frac{n^{2}}{n^{2}-1})^{n}*\frac{n}{n+1}=
(\frac{n^2-1+1}{n^{2}-1})^{n}\frac{n}{n+1}=
(1+\frac{1}{n^{2}-1})^{n}\frac{n}{n+1}>(1+\frac{n}{n^{2}-1})\frac{n}{n+1}>(1+\frac{n}{n^{2}})\frac{n}{n+1}=1[/tex]

hence [tex]t_n-_1>t_n[/tex] which means that the sequence is decreasing...


Try to show some work of yours, for people here won't do your homework, and this way you may get more replies.


P.S. This looks more like calculus..lol...where does real analysis come into play here?!
 
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1. What is the purpose of Real Analysis Proof?

Real Analysis Proof is a branch of mathematics that deals with the rigorous study of real numbers and their properties. Its purpose is to provide a foundation for calculus and other mathematical concepts by defining and proving the properties and relationships of real numbers.

2. What are r(n), t(n), e, and n in Real Analysis Proof?

In Real Analysis Proof, r(n) refers to the remainder term in the Taylor series expansion of a function, t(n) refers to the nth term in a sequence, e refers to the mathematical constant representing the base of the natural logarithm, and n refers to a variable or parameter in a mathematical expression or equation.

3. How does Real Analysis Proof differ from other branches of mathematics?

Real Analysis Proof is more focused on rigorous proofs and theoretical concepts, while other branches of mathematics may focus more on applications and computations. It also uses a specific set of axioms and definitions to define and prove the properties of real numbers.

4. What are some common techniques used in Real Analysis Proof?

Some common techniques used in Real Analysis Proof include proof by contradiction, induction, and direct proof. Other techniques such as limit theorems, continuity, and differentiation also play important roles in the field.

5. How can one improve their skills in Real Analysis Proof?

To improve skills in Real Analysis Proof, it is important to practice regularly and work through a variety of problems and proofs. It can also be helpful to study and understand the underlying concepts and theories behind the proofs, rather than just memorizing them. Seeking guidance from experienced mathematicians or studying textbooks and online resources can also be beneficial.

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