# DiscreteMaxLimit

DiscreteMaxLimit[f,k]

gives the max limit kf(k) of the sequence f as k tends to over the integers.

DiscreteMaxLimit[f,{k1 ,,kn }]

gives the nested max limit  f(k1,,kn) over the integers.

DiscreteMaxLimit[f,{k1,,kn}{ ,, }]

gives the multivariate max limit f(k1,,kn) over the integers.

# Details and Options   • DiscreteMaxLimit is also known as limit superior, supremum limit, limsup, upper limit and outer limit.
• DiscreteMaxLimit computes the smallest upper bound for the limit and is always defined for real-valued sequences. It is often used to give conditions of convergence and other asymptotic properties that do not rely on an actual limit to exist.
• DiscreteMaxLimit[f,k] can be entered as f. A template can be entered as dMlim , and moves the cursor from the underscript to the body.
• DiscreteMaxLimit[f,{k1,,kn}{ ,, }] can be entered as  f.
• The possible limit points are ±.
• The max limit is defined as a limit of the max envelope sequence max[ω]:
•  DiscreteMaxLimit[f,k∞] DiscreteLimit[max[ω],ω∞] DiscreteMaxLimit[f,{k1,…,kn}{∞,…,∞}] DiscreteLimit[max[ω],ω∞]
• DiscreteMaxLimit[f[k],k-] is equivalent to DiscreteMaxLimit[f[-l],l] etc.
• The definition uses the max envelope max[ω]MaxValue[{f[k],kωk },k] for univariate f[k] and max[ω]MaxValue[{f[k1,,kn],k1ωknωki },{k1,,kn}] for multivariate f[k1,,kn]. The sequence max[ω] is monotone decreasing as ω, so it always has a limit, which may be ±.
• The illustration shows max[k] and max[Min[k1,k2]] in blue.
• • DiscreteMaxLimit returns unevaluated when the max limit cannot be found.
• The following options can be given:
•  Assumptions \$Assumptions assumptions on parameters GenerateConditions Automatic whether to generate conditions on parameters Method Automatic method to use PerformanceGoal "Quality" aspects of performance to optimize
• Possible settings for GenerateConditions include:
•  Automatic non-generic conditions only True all conditions False no conditions None return unevaluated if conditions are needed
• Possible settings for PerformanceGoal include \$PerformanceGoal, "Quality" and "Speed". With the "Quality" setting, DiscreteMaxLimit typically solves more problems or produces simpler results, but it potentially uses more time and memory.

# Examples

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## Basic Examples(4)

Max limit of a sequence:

Max limit of a product:

Use dMlim to enter the template and to move from the underscript to the body:

## Scope(21)

### Basic Uses(4)

Compute the max limit of a sequence when n approaches Infinity:

Compute the max limit of a sequence when n approaches :

Compute a nested max limit for a multivariate sequence:

Compute the max limit of a list of sequences:

### Elementary Sequences(6)

Find the max limit of a rational-exponential sequence:

Convergent geometric sequence:

Oscillating geometric sequence:

Divergent oscillating geometric sequence:

Exponential sequence:

Power sequence:

Trigonometric sequences:

Inverse trigonometric sequence:

Logarithmic sequence:

### Periodic Sequences(3)

Limits of periodic sequences:

Eventually periodic sequence:

Densely aperiodic sequences:

### Piecewise Sequences(2)

Piecewise sequence with a finite max limit:

Piecewise sequence with an infinite max limit:

Piecewise sequence with periodic conditions:

### Special Function Sequences(2)

Compute the limit of a sequence involving Fibonacci:

Sequence involving FactorialPower:

### Number Theoretic Sequences(2)

Limits involving LCM and GCD:

Sequence involving Prime:

### Multivariate Sequences(2)

Compute a nested max limit:

Plot the sequence and its limit:

Multivariate max limits:

## Options(6)

### Assumptions(1)

Specify assumptions on a parameter:

Different assumptions can produce different results:

### GenerateConditions(3)

Return a result without stating conditions:

This result is only valid if x>1:

Return unevaluated if the results depend on the value of parameters:

By default, conditions are generated that return a unique result:

By default, conditions are not generated if only special values invalidate the result:

With , even these non-generic conditions are reported:

### Method(1)

Compute the max limit of a periodic sequence using the default method:

Obtain the same answer using the method for periodic sequences:

The limit of the sequence is undefined, since it oscillates between 0 and 1:

### PerformanceGoal(1)

DiscreteMaxLimit computes limits involving sequences of arbitrarily large periods:

Use PerformanceGoal to avoid potentially expensive computations in such cases:

The Method option overrides PerformanceGoal:

## Applications(7)

### Basic Applications(2)

Compute the asymptotic supremum of a sequence:

Plot the sequence and the asymptotic supremum:

Verify that the following sequence does not have a limit:

Show that DiscreteMaxLimit and DiscreteMinLimit are not equal:

Confirm that the limit does not exist by using DiscreteLimit:

### Series Convergence(4)

Show that the infinite series whose general term is defined here is convergent, by using the ratio test:

Plot the partial sums of the series:

Compute the ratio of the adjacent terms using DiscreteRatio:

The sequence of ratios does not converge:

However, the ratio test can still be used because the upper limit of the ratios is less than 1:

Confirm that the series converges using SumConvergence:

Evaluate the infinite series:

Show that the infinite series whose general term is defined here is convergent, by using the root test:

Plot the partial sums of the series:

Compute the n root of the general term:

The limit of the sequence of roots does not exist:

However, the root test still indicates convergence because the max limit is less than 1:

Confirm that the series converges using SumConvergence:

Evaluate the infinite series:

Consider the sequence :

The inverse radius of the associated power series is given by:

This means the radius of convergence is infinite and converges for all , in particular to :

Compute the Taylor series at zero and its radius of convergence for the following function:

The  Taylor coefficient is :

Formally, the Taylor series does sum to the original function:

The radius of convergence of the Taylor series is given by:

This means the Taylor series will converge for values of within of the origin. For example, at :

At values of further away, the sum will not converge; for example, at :

At the points , the terms of the Taylor series alternate between and :

Hence the partial sums go between and :

Visualize and the partial sums of its Taylor series on the interval ; in the interior of the interval, convergence is rapid, but the Taylor polynomials always go to either or at the endpoints:

### Computational Complexity(1)

An algorithm runtime function is said to be "big-o of ", written , if :

Similarly, is said to be "big-theta of ", written if and :

The statement is always true:

If and , then :

It is possible for two functions to share neither relationship:

Hence, defines a reflexive partial order on the space of algorithm runtimes similar to :

If and , then , which implies that is an equivalence relation:

## Properties & Relations(11)

A real-valued sequence always has a (possibly infinite) max limit:

The corresponding limit may not exist:

If and have finite max limits, then :

In this case, there is strict inequality:

Positive multiplicative constants can be moved outside a limit:

For a real-valued sequence, if DiscreteLimit exists, DiscreteMaxLimit has the same value:

If has a finite limit, then :

DiscreteMaxLimit is always greater than or equal to DiscreteMinLimit:

If DiscreteMaxLimit equals DiscreteMinLimit, the limit exists and equals their common value:

If the max limit is , then the min limit and thus the limit are also :

DiscreteMaxLimit can be computed as -DiscreteMinLimit[-f,]:

If , then :

If the two max limits are equalas in this examplethen has a limit:

This is a generalization of the "squeezing" or "sandwich" theorem:

MaxLimit is always greater than or equal to DiscreteMaxLimit:

## Possible Issues(1)

DiscreteMaxLimit is only defined for real-valued sequences: ## Neat Examples(1)

Visualize a set of sequence max limits:

Introduced in 2017
(11.2)