Compute the limit of a sequence when n approaches Infinity:

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

Compute a nested limit for a multivariate sequence:

Compute the limit of a list of sequences:

Find the limit of a rational sequence:

Geometric sequence:

Exponential sequences:

Trigonometric sequence:

Inverse trigonometric sequence:

Logarithmic sequence:

Find the limit of ArcTan[Log[n]]:

Compute the limit of a binomial sequence:

Limits of sequences involving FactorialPower:

Limits of sequences involving Factorial:

Compute limits involving Fibonacci and LucasL:

Limit involving Pochhammer:

Convergent alternating sequence:

Divergent alternating sequence:

Oscillatory alternating sequence:

Limits involving periodic sequences:

Eventually periodic sequence:

Densely aperiodic sequences:

A convergent piecewise sequence:

A divergent piecewise sequence:

Piecewise sequence with periodic conditions:

Limit involving Floor:

Compute limits involving Prime:

Prime is of order :

Limits involving PrimePi:

PrimePi is of order :

Limits involving PartitionsP and PartitionsQ:

Limits involving other number theoretic sequences:

Compute a nested sequence limit:

Plot the sequence and its limit:

Multivariate sequence limits:

Compute limits of sequences involving Inactive sums:

Nested limit of an Inactive sum:

Obtain the same result in two steps using an interchange of DiscreteLimit and Sum:

Limits of sequences involving Inactive products:

Nested limit of an Inactive product:

Obtain the same result in two steps using an interchange of DiscreteLimit and Product:

Limits of sequences involving Inactive continued fractions:

Nested limit of an Inactive continued fraction:

Obtain the same result in two steps using DiscreteLimit and ContinuedFractionK:

Specify assumptions on a parameter:

Different assumptions can produce different results:

Return a result without stating conditions:

This result is only valid if y>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 GenerateConditions->True, even these non-generic conditions are reported:

Compute the limit of a sequence using the default method:

Obtain the same answer using a call to Limit:

The given sequence is not periodic, hence the method for periodic sequences fails:

DiscreteLimit computes limits involving sequences of arbitrarily large periods:

Use PerformanceGoal to avoid potentially expensive computations in such cases:

The Method option overrides PerformanceGoal:

The perimeter of a regular polygon of radius r and n sides:

In the limit n->∞, the perimeter approaches the circumference of a circle of radius r:

The area of a regular polygon of radius r and n sides:

In the limit n->∞, this approaches the area of a circle of radius r:

Visualize the inscribed polygon and the approximate perimeter and area as n increases:

Consider covering a ball of radius r by 2n cylinders as shown in the figure:

The volume of the cylinders is:

Taking the DiscreteLimit as n->Infinity gives the volume of the ball:

Compare with a direct computation:

Consider the following function and a set of rectangles defined by its plot:

For n5 on the interval [0,2], the rectangles are the following:

The area of these rectangles defines a Riemann sum that approximates the area under the curve:

Use DiscreteLimit to obtain the exact answer:

Obtain the same area directly using Integrate:

Visualize the process for this function as well as three others:

Compute an infinite sum as the limit of a finite sum:

Obtain the same answer using Sum:

The following sequence defines a convergent series:

Find the value of the series:

Compute the result directly using Sum:

Prove that an infinite series is divergent, starting with the sum of a finite number of terms:

The series diverges, since the limit of the finite sums does not exist:

Confirm the divergence using SumConvergence and Sum:

Obtain the Abel sum of the series using Regularization:

Compute a doubly infinite sum as a nested limit of a finite sum:

Obtain the same answer directly using Sum:

Compute an infinite product as a limit of a finite product:

Obtain the same answer using Product:

Construct a rotation matrix as a limit of repeated infinitesimal transformations:

Compare with a direction construction:

Use the ratio test to verify convergence of a series whose general term is given by:

Compute the DiscreteRatio for this series:

The series converges, since the limit of the ratio is less than 1:

Verify the result using SumConvergence:

Use the root test to verify convergence of a series whose general term is given by:

The series converges, since the limit of the n root is less than 1:

Verify the result using SumConvergence:

Use the Raabe test to verify convergence of a series whose general term is given by:

Raabe's test applies because the ratio test is inconclusive:

The series converges, since the following limit is greater than 1:

Verify the result using SumConvergence:

Use the divergence test to verify divergence of a series whose general term is given by:

The series diverges, since the limit of the general term is not 0:

Verify the result using SumConvergence:

Show that the following sequence converges to 0, and verify the classical definition with ϵ=1/7:

Compute the limit:

Set the value of :

Use Reduce to show that the definition is satisfied for all n>=12:

Verify the result using DiscretePlot:

Show that the following sequence diverges to Infinity, and verify the classical definition with M=35:

Compute the limit:

Set the value of M:

Use Reduce to show that the definition is satisfied for all n >= 10:

Verify the result using DiscretePlot:

Determine the convergence of the harmonic series , whose terms are given by:

Standard tests such as the ratio test are inconclusive:

Define an auxiliary series as follows:

The terms of consist of runs of length of :

Notice that :

Also, the sum of each run is , so the sum of the first terms is :

The partial sums of are called the harmonic numbers :

For any positive integer , , so eventually exceeds and diverges to :

This means the sum of does not converge:

The divergence is slow, however, requiring more than terms just to get over :

Compute the limit of a nonlinear recursive sequence that is specified using RSolveValue:

Compute the limit of a trigonometric recursive sequence that is specified using RSolveValue:

Compute the value of :

Compute as the limit of a sequence:

Compute as the limit of a Sum:

Compute as the limit of a sequence:

Compute EulerGamma using the limit of a sequence:

Compute the golden ratio using a sequence involving Fibonacci:

Represent as the limit of a sequence with symbolic entries:

Represent Log[x] as the limit of a sequence:

The Stolz–Cesàro theorem is a discrete version of L'Hôpital's rule, and can be used to compute the limits for ratios of sequences, under suitable conditions. The theorem states that:

Verify the Stolz–Cesàro theorem for the sequences defined by:

Compute the limit for the ratio of differences:

Obtain the same result directly using DiscreteLimit:

Plot the sequence and the limit:

Verify the Stolz–Cesàro theorem for the sequences defined by:

Compute the limit for the ratio of differences:

Obtain the same result directly using DiscreteLimit:

Plot the sequence and the limit:

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

Similarly, is said to be "little-omega of ", written , if :

If , then :

It is possible for two functions to share neither relationship:

Moreover, neither relationship even holds between a function and itself:

Hence, and define partial orders on the space of algorithm runtimes:

if the algorithm associated to is much faster than the one associated to for large inputs:

denotes the opposite relationship:

Note that the two lists are not exactly reversed, because and are incomparable:

An algorithm runtime function is said to be "big-theta of ", written , if the following holds:

Consider an algorithm that takes time —a polynomial of degree —to run:

The ratio of this function to the monomial goes to the leading coefficient at infinity:

Since the limit of the sequence exists, its max and min limits must both equal this value:

For an algorithmic runtime, must be a positive finite number, so every polynomial algorithm is :

Hence, only the leading term in the polynomial is important in determining the runtime for large inputs:

Check the asymptotic complexity of the fast Fourier transform:

Compute the asymptotic complexity: