# I cant understand this prove explanation on limit series

• transgalactic
In summary, the conversation discusses the proof of a continues function f(x) that is bounded on (x_0,+infinity). The proof shows that for every T, there exists a sequence X_n=+infinity such that lim [f(x_n +T) - f(x_n)]=0 as n->+infinity. The conversation also mentions the use of delta in limit proofs for finite numbers and the use of a large number M in limit proofs for approaching infinity.
transgalactic
there is a continues function f(x) and bounded on (x_0,+infinity)
proove that for every T there is a sequence
X_n=+infinity
so
lim [f(x_n +T) - f(x_n)]=0
n->+infinity

i was told:
uppose that $\lim_{x\to\infty}f(x)=a$. So we know that given any $\varepsilon>0$ there exists a $\eta>0$ such that $\eta< x\implies |f(x)-a|<\varepsilon~(1)$, and since $x_n\to \infty$ we may find a $\delta>0$ such that $\delta<n\implies \eta<x_n~(2)$. Now suppose that $T>0$ (the proof for the other cases is analgous), then choose $\delta$ such that $(2)\implies (1)$ then $\left[f\left(x+T\right)-f(x)\right|\leqslant \left|f\left(x+T\right)-a\right|+|f(x)-a|<2\varepsilon$, this implies the result.

http://www.mathhelpforum.com/math-h...e-never-learnt-well-epsilon-delta-proofs.html

i learned from the delta proofes article that when you define the delta

$\delta>0$
it needs to come with
$|x-x_3|<\delta$
in our case x_3 goes to infinity
so the inqueality that i presented not logical

but on the other hand
it how its done on the article limit proove

??

Limits in which x approaches infinity are proved differently than those for which x approaches some finite number a.

$$\lim_{x \rightarrow \infty} f(x) = L$$

means that for any $\epsilon > 0$ there exists a number M > 0, such that for any x > M, then |f(x) - L| < $\epsilon$

If I want to prove the limit above to you, you tell me how close f(x) has to be to L (you give me $\epsilon$), and I tell you a number M.

If you're not satisfied, you tell me another $\epsilon$ that's even smaller, and I have to find another M (even larger).

And so on, until you're convinced that I can force f(x) as close to L as you like, by specifiying how big x has to be.

Got it?

in your explanation M is delta>0
??

transgalactic said:
in your explanation M is delta>0
??
No. M is generally a pretty large number, while $\delta$ is usually very small. A big difference is the definition I showed doesn't try to get x within $\delta$ of infinity.

but he does use delta

for what purpose
??

If the limit is as x approaches a finite number a, you use $\delta$, since you want to make x very close to a. I.e., you want to make |x - a| < $\delta$.
If the limit is as x approaches infinity, that's a different matter, since as I explained earlier, you can't get x within $\delta$ of infinity. You can, however, make x larger than some (presumably large) number M.

I don't think I can make it any clearer than that.

## What is a limit series?

A limit series is a mathematical sequence of numbers where the terms approach a specific value, known as the limit, as the number of terms increases. It is commonly used to analyze the behavior of functions and determine their convergence or divergence.

## Why is it important to understand limit series?

Limit series are important in mathematics because they are used to prove the convergence or divergence of various mathematical functions and series. They also have applications in fields such as physics, engineering, and economics.

## How do you prove a limit series?

To prove a limit series, you must show that the terms of the series approach a specific value, known as the limit, as the number of terms increases. This can be done using various mathematical techniques such as the limit comparison test, ratio test, or root test.

## What are some common challenges in understanding limit series?

Some common challenges in understanding limit series include the complexity of the mathematical concepts involved, the use of abstract symbols, and the need for a strong foundation in calculus and algebra. It can also be challenging to apply the various tests and techniques to different types of limit series.

## What are some resources for learning more about limit series?

There are many resources available for learning about limit series, including textbooks, online tutorials, and video lectures. You can also consult with a math teacher or tutor for personalized help. Additionally, practicing problems and working through examples can greatly aid in understanding limit series.

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