Convergence of Series: Proving the Relationship Between Two Converging Series

In summary, the instructor assigned a problem from Rudin's Principles of Mathematical Analysis; a problem which I have been unable to solve after giving it good thought. However, by reviewing the ideas of chapter one, I was able to find a solution.
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Our instructor assigned a problem from Rudin's Principles of Mathematical Analysis; a problem which I have been unable to solve after giving it good thought.

The statement is:
"Prove that the convergence of SUM[an] implies the convergence of SUM[sqrt(an)/n], if an >= 0."

The instructor did give us a hint: "Review the ideas of chapter one," from which I gleaned the archimedean property or supremums might be important. Anyway, if anyone is familiar with how this can be proved, I would appreciate a nudge in the right direction. Thanks.
 
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  • #2
In case anyone else read this, I have found the solution on the internet. It was embarassingly simple. By the AM-GM inequality:

[tex]a_n + \frac{1}{n^2} \ge 2\frac{\sqrt{a_n}}{n}.[/tex]​

The left hand series converges, so by direct comparison, the right hand series also converges.
 
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  • #3
Cool, I didn't know of this inequality. What does AM-GM stands for?
 
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Harmonic - Geometric - Arithmetic Mean Inequality

AM-GM = Arithmetic Mean - Geometric Mean

In full, the Harmonic - Geometric - Arithmetic Mean inequality is (for sequences)

[tex]\frac{n}{\frac{1}{a_{1}}+\frac{1}{a_{2}}+\cdot\cdot\cdot +\frac{1}{a_{n}}}\leq \left( a_{1}a_{2}\cdot\cdot\cdot a_{n}\right) ^{\frac{1}{n}}\leq \frac{a_{1}+a_{2}\cdot\cdot\cdot + a_{n}}{n}[/tex]

where equality holds iff the [itex]a_{i}[/itex]'s are all equal, and it is understood that [itex]a_{i}\geq 0,\forall i[/itex].
 
  • #5
Harmonic - Geometric - Arithmetic Mean inequality is (for functions)

Suppose that the real-valued function [tex]f(x)[/tex] is defined, properly integrable, and strictly positive on the interval [tex] \left[ x_{1}, x_{2}\right] [/tex].

Then the Harmonic - Geometric - Arithmetic Mean inequality is (for functions)

[tex] \frac{x_{1} - x_{2}}{\int_{x_{1}}^{x_{2}} \frac{dx}{f(x)}} \leq \exp\left( \frac{1}{x_{1} - x_{2}}\int_{x_{1}}^{x_{2}} \log f(x) dx\right) \leq \frac{1}{x_{1} - x_{2}} \int_{x_{1}}^{x_{2}} f(x)dx[/tex]
 
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  • #6
Great, I'm noting all of this down.
 

FAQ: Convergence of Series: Proving the Relationship Between Two Converging Series

How do you define the convergence of a series?

The convergence of a series is determined by whether the sum of its terms approaches a finite value as the number of terms increases, or diverges to infinity. This is often denoted by the limit of the series, which is calculated by taking the limit of the sequence of partial sums.

What is the relationship between two converging series?

The relationship between two converging series is that they both approach a finite sum as the number of terms increases. This means that the limit of their partial sums will also be finite and equal.

How do you prove the relationship between two converging series?

The most common way to prove the relationship between two converging series is by using the comparison test or the limit comparison test. These tests involve comparing the given series to a known converging or diverging series and using their properties to determine the convergence or divergence of the original series.

What is the importance of proving the relationship between two converging series?

Proving the relationship between two converging series is important because it allows us to determine the convergence or divergence of a series by comparing it to a known series. This helps us understand the behavior of a series and its sum, which can have practical applications in various fields of science and mathematics.

Are there any other methods to prove the relationship between two converging series?

Yes, there are other methods to prove the relationship between two converging series, such as the ratio test, the root test, and the integral test. These tests also involve comparing the given series to a known series and using their properties to determine the convergence or divergence of the original series.

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