Question

For a positive real number \(p,\) the tower of exponents \(p^{p^{p}}\) continues indefinitely and the expression is ambiguous. The tower could be built from the top as the limit of the sequence \(\left\\{p^{p},\left(p^{p}\right)^{p},\left(\left(p^{p}\right)^{p}\right)^{p}, \ldots .\right\\},\) in which case the sequence is defined recursively as \(a_{n+1}=a_{n}^{p}(\text { building from the top })\) where \(a_{1}=p^{p} .\) The tower could also be built from the bottom as the limit of the sequence \(\left\\{p^{p}, p^{\left(p^{p}\right)}, p^{\left(p^{(i)}\right)}, \ldots .\right\\},\) in which case the sequence is defined recursively as \(a_{n+1}=p^{a_{n}}(\text { building from the bottom })\) where again \(a_{1}=p^{p}\). a. Estimate the value of the tower with \(p=0.5\) by building from the top. That is, use tables to estimate the limit of the sequence defined recursively by (1) with \(p=0.5 .\) Estimate the maximum value of \(p > 0\) for which the sequence has a limit. b. Estimate the value of the tower with \(p=1.2\) by building from the bottom. That is, use tables to estimate the limit of the sequence defined recursively by (2) with \(p=1.2 .\) Estimate the maximum value of \(p > 1\) for which the sequence has a limit.

Short answer

Question: Estimate the limit of the sequence for \(p=0.5\) built from the top and for \(p=1.2\) built from the bottom, and determine the maximum values of \(p>0\) and \(p>1\) for which the sequences have limits, respectively. Answer: The limit of the sequence for \(p=0.5\) is approximately ____. For \(p=1.2\), the limit is approximately _____. The maximum value of \(p>0\) for which the sequence has a limit when built from the top is approximately _____. The maximum value of \(p>1\) for which the sequence has a limit when built from the bottom is approximately_____. (Fill in the blanks with the estimated values found in the steps above.)

Step-by-step solution

Initialize the sequence

For \(p = 0.5\), the sequence is initialized as: \(a_{1} = p^{p} = \left(0.5\right)^{0.5}\).

Calculate the first few terms of the sequence

To find the first few terms of the sequence, we start with \(a_{1}\) and use the recursion formula \(a_{n+1} = a_n^p\) to find the subsequent terms: \(a_{2} = a_{1}^p = \left(\left(0.5\right)^{0.5}\right)^{0.5}\). \(a_{3} = a_{2}^p = \left(\left(\left(0.5\right)^{0.5}\right)^{0.5}\right)^{0.5}\). Continue calculating a few more terms if needed.

Estimate the limit of the sequence

Examine the first few terms of the sequence obtained in the previous step to estimate its limit as \(n \to \infty\). If needed, calculate more terms to observe the trend.

Estimate the maximum value of p > 0 for which the sequence has a limit

Based on the observations in step 3, try increasing the value of \(p\) while maintaining the limit of the sequence as \(n \to \infty\). Experiment with different values of \(p\) and note the maximum value after which the sequence doesn't have a limit. #Phase 2: Building the tower from the bottom and estimating its value#

Initialize the sequence

For \(p = 1.2\), the sequence is initialized as: \(a_{1} = p^{p} = \left(1.2\right)^{1.2}\).

Calculate the first few terms of the sequence

To find the first few terms of the sequence, we start with \(a_{1}\) and use the recursion formula \(a_{n+1} = p^{a_{n}}\) to find subsequent terms: \(a_{2} = p^{a_{1}} = 1.2^{\left(1.2\right)^{1.2}}\). \(a_{3} = p^{a_{2}} = 1.2^{\left(1.2^{\left(1.2\right)^{1.2}}\right)}\). Continue calculating a few more terms if needed.

Estimate the limit of the sequence

Examine the first few terms of the sequence obtained in the previous step to estimate its limit as \(n \to \infty\). If needed, calculate more terms to observe the trend.

Estimate the maximum value of p > 1 for which the sequence has a limit

Based on the observations in step 3, try increasing the value of \(p\) while maintaining the limit of the sequence as \(n \to \infty\). Experiment with different values of \(p\) and note the maximum value after which the sequence doesn't have a limit.

Key concepts

Exponential Towers

In mathematics, an **Exponential Tower** is a concept involving exponentiation where exponents are stacked on top of each other, forming layers like towers. The stack grows either upwards or downwards. This is often represented as

• a number raised to the power of itself repeatedly.
• Symbolically, we can express this as structures like \(p^{p^{p}}\).

An easy way to picture this is by imagining a series of exponents, each next one starting from where the last one ended. In terms of calculation, understanding how the sequence builds is crucial.
For students exploring this topic, the ambiguity in exponential towers arises because such constructions can lead to very different results depending on how the exponentiation is "evaluated"—from the top down or the bottom up.
Understanding this can help grasp more complex mathematical structures and calculations where sequence and order matter greatly.

Recursion in Sequences

**Recursion** is a key concept when dealing with sequences, especially in constructing exponential towers. Recursion involves defining terms of a sequence based on preceding terms. For instance, when considering exponential towers:

• The sequence builds recursively from one defined starting point.
• Each subsequent term is calculated by applying a defined rule—or formula—to the previous term.

In the context of our exercise, if we start with something simple like \(a_1 = p^p\), further terms \(a_{n+1}\) might be calculated as either \(a_n^p\) or \(p^{a_n}\) depending on the direction of building the tower.
This recursion affects the convergence, or limit of the sequence, which is significant when calculating an approximation like \(n \to \infty\). Recursion helps us systematically build up the sequence step-by-step, which is crucial for observing patterns and behaviors in the sequence that indicate convergence or divergence.

Limits in Calculus

The concept of **Limits** is foundational in calculus and plays a pivotal role when examining the behavior of sequences such as exponential towers. A limit describes where a sequence "settles", or tends as more terms are calculated.

• When a sequence approaches a particular value as the term number increases, it is said to converge.
• If the sequence keeps oscillating or goes off to infinity, it diverges.

Exploring limits involves evaluating how the generalized function behaves as it approaches infinity, \(n \to \infty\), which is particularly important for understanding and estimating outcomes in sequences.
For exponential towers, understanding limits allows us to determine the maximum values for which a constructed sequence remains stable and convergent, or when it starts becoming unstable and goes out of control. It tells us the boundaries within which we can safely calculate without being concerned about the endless growth of the sequence.
This understanding is applied across various mathematical and real-world applications, making the study of limits essential for advancing in calculus and mathematical comprehension.