# Infinite Series .... Sohrab Exercise 2.3.10 (1) .... ....

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In summary: in summary, the author provides a formal and rigorous proof that the series $\sum_{n=1}^\infty x_n$ is convergent.
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I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition).

I am focused on Chapter 2: Sequences and Series of Real Numbers ... ...

I need help In order to formulate a rigorous proof to the proposition stated in Exercise 2.3.10 (1) ... ...

Exercise 2.3.10 (1) reads as follows:
View attachment 9062I am unsure of what would constitute a formal and rigorous proof to the proposition or statement of Exercise 2.3.10 (1) ...

Here is my attempt at a formal and rigorous proof ...Assume $$\displaystyle \sum_{ n = 1 }^{ \infty } x_n$$ is convergent ... then $$\displaystyle \sum_{ n = 1 }^{ \infty } x_n$$ is Cauchy ...Thus we have the following ...... for every $$\displaystyle \epsilon \gt 0 \ \exists \ N \text{ such that } m_0 \geq n_0 \geq N \ \Longrightarrow \ \vert s_{ m_0 } - s_{ n_0 } \vert \lt \epsilon$$ ... ...... that is ... for every $$\displaystyle \epsilon \gt 0 \ \exists \ N \text{ such that } m_0 \geq n_0 \geq N \ \Longrightarrow \ \left\vert \sum_{ k = n_0 + 1 }^{ m_0 } x_n \right \vert \lt \epsilon$$ ... ...If $$\displaystyle n_0 \geq m$$ then we are done ...If $$\displaystyle n_0 \lt m_0$$ ... then take $$\displaystyle n_0 = m$$ and we are done ...Now assume $$\displaystyle \sum_{ n = m }^{ \infty } x_n$$ is convergent ... ... then ... for every $$\displaystyle \epsilon \gt 0 \ \exists \ N \text{ such that } m_0 \geq n_0 \geq m \geq N \ \Longrightarrow \ \left\vert \sum_{ k = n_0 + 1 }^{ m_0 } x_n \right \vert \lt \epsilon$$ ... ...Clearly the above N ensures that for $$\displaystyle \sum_{ n = 1 }^{ \infty } x_n$$ ...

... $$\displaystyle \epsilon \gt 0 \ \exists \ N \text{ such that } m_0 \geq n_0 \geq N \ \Longrightarrow \ \left\vert \sum_{ k = n_0 + 1 }^{ m_0 } x_n \right \vert \lt \epsilon$$ ... ...

... that is $$\displaystyle \sum_{ n = 1 }^{ \infty }x_n$$ is convergent ...

Could someone please indicate whether the above proof is correct and acceptable ,,, indeed ... have I missed the point ...

Further could someone critique the above proof pointing out errors and deficiencies ...Help will be appreciated ...

Peter

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$\displaystyle\sum_{n=1}^\infty x_n$ is convergent​

$\displaystyle\iff\ \lim_{k\to\infty}\sum_{n=1}^k x_n$ exists and is finite

$\displaystyle\iff\ \lim_{k\to\infty}\sum_{n=m}^k x_n$ $=$ $\displaystyle\lim_{k\to\infty}\left[\left(\sum_{n=1}^k x_n\right)-\sum_{n=1}^{m-1}x_n\right]$ $=$ $\displaystyle\left[\lim_{k\to\infty}\left(\sum_{n=1}^k x_n\right)\right]-\sum_{n=1}^{m-1}x_n$ exists and is finite

$\displaystyle\iff\ \sum_{n=m}^\infty x_n$ is convergent.

In other words, adding a finite number of terms to or subtracting a finite number of terms from a convergent series results in convergent series.

Olinguito said:
$\displaystyle\sum_{n=1}^\infty x_n$ is convergent​

$\displaystyle\iff\ \lim_{k\to\infty}\sum_{n=1}^k x_n$ exists and is finite

$\displaystyle\iff\ \lim_{k\to\infty}\sum_{n=m}^k x_n$ $=$ $\displaystyle\lim_{k\to\infty}\left[\left(\sum_{n=1}^k x_n\right)-\sum_{n=1}^{m-1}x_n\right]$ $=$ $\displaystyle\left[\lim_{k\to\infty}\left(\sum_{n=1}^k x_n\right)\right]-\sum_{n=1}^{m-1}x_n$ exists and is finite

$\displaystyle\iff\ \sum_{n=m}^\infty x_n$ is convergent.

In other words, adding a finite number of terms to or subtracting a finite number of terms from a convergent series results in convergent series.
Thanks for the help Olinguito ... neat clear proof!

Peter

## 1. What is an infinite series?

An infinite series is a sum of an infinite number of terms. It is written in the form of a1 + a2 + a3 + ..., where an is the nth term of the series.

## 2. What is the difference between a finite and infinite series?

A finite series has a limited number of terms, while an infinite series has an unlimited number of terms.

## 3. How do you determine if an infinite series converges or diverges?

To determine if an infinite series converges or diverges, you can use various tests such as the comparison test, ratio test, root test, or integral test. These tests help determine if the terms of the series approach a finite limit or if they continue to increase without bound.

## 4. What is the significance of the convergence of an infinite series?

The convergence of an infinite series indicates that the sum of the terms in the series approaches a finite value. This is important in many mathematical and scientific applications, as it allows for calculations and predictions to be made with a high degree of accuracy.

## 5. How are infinite series used in real-world applications?

Infinite series are used in many fields of science and mathematics, such as physics, engineering, and finance. They are used to model and predict natural phenomena, calculate probabilities, and solve complex mathematical problems. For example, the concept of infinite series is used in calculus to find the area under a curve, which has many practical applications in fields like physics and economics.

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