# Question on the Cauchy Condensation test

• Simfish
In summary, the Cauchy condensation test is a method used to estimate the convergence of an infinite series by forming summation blocks and finding a useful estimate for each block. The value of the base 2 in 2^k is not important, as it is the exponential that is crucial. The function f in the series only needs to be well-defined for the natural numbers. The sequence in the series is positive, monotone decreasing, and denoted by f(n) or a_n. The biggest summand in each block is the first one and the goal is to find a useful estimate for each block.
Simfish
Gold Member
So it's here
http://en.wikipedia.org/wiki/Cauchy_condensation_test

My question is this: is the value of the base 2 in 2^k an arbitrary value? Or is there something special about 2? Can we just use something like e^k instead?

The 2 is not important, it's the exponential that is crucial.

Simfishy said:
So it's here
http://en.wikipedia.org/wiki/Cauchy_condensation_test

My question is this: is the value of the base 2 in 2^k an arbitrary value? Or is there something special about 2? Can we just use something like e^k instead?

Notice that the function f in $\sum_{n=0}^{\infty} 2^{n}f(2^{n})$ need not be well-defined for all arguments in the real numbers, but only for the natural numbers (including the zero).
What you have is a positive monotone decreasing sequence (which is denoted here by $f(n)$ but could just as well be written as $a_n$). Now consider the sum (infinite series): the idea now is to form summation blocks with length $2^n$ and find a useful estimate for each block. Every block incorporates $2^n$ summands and the biggest summand in each block is the first one, as the sequence is monotone decreasing.

Try writing this down and see what you end up with. If you can, check the converse.

## 1. What is the Cauchy Condensation test?

The Cauchy Condensation test is a convergence test used in mathematical analysis to determine the convergence or divergence of an infinite series. It was developed by the French mathematician Augustin-Louis Cauchy in the 19th century.

## 2. How does the Cauchy Condensation test work?

The Cauchy Condensation test is based on the idea that the convergence of a series is not affected by multiplying each term by a constant. It involves creating a new series by multiplying each term in the original series by a power of 2, and then comparing the convergence of the two series.

## 3. What is the formula for the Cauchy Condensation test?

The formula for the Cauchy Condensation test is: if the terms of a series are positive and decreasing, then the series converges if and only if the series formed by multiplying each term by a power of 2 also converges.

## 4. What is the significance of the Cauchy Condensation test?

The Cauchy Condensation test is significant because it provides a simple and efficient method for determining the convergence or divergence of a series. It is particularly useful for series with rapidly decreasing terms, as it can quickly determine their convergence.

## 5. What are some examples of series where the Cauchy Condensation test is used?

The Cauchy Condensation test is commonly used in the analysis of infinite series in calculus and real analysis. Some examples of series where it is used include the geometric series, the harmonic series, and the series for the natural logarithm function.

• Calculus
Replies
1
Views
2K
• Calculus
Replies
2
Views
3K
• Calculus
Replies
24
Views
3K
• Calculus
Replies
15
Views
3K
• Calculus
Replies
4
Views
4K
• Calculus
Replies
4
Views
1K
• DIY Projects
Replies
6
Views
1K
• Calculus
Replies
8
Views
2K
• Calculus
Replies
3
Views
2K
• Materials and Chemical Engineering
Replies
18
Views
1K