A definite sum over a finite range:

Using step size 2:

Using a finite list of elements:

Plot the sequence and its partial (or cumulative) sums:

A multiple sum over finite ranges:

Using a different step size:

Plot a multivariate sequence and its partial sums:

The outermost summation bounds can depend on inner variables:

Combine summation over lists with standard iteration ranges:

The elements in the iterator list can be any expression:

Sum over infinite ranges:

Multivariate sums over infinite ranges:

Sum over symbolic ranges:

Indefinite summation:

The difference is equivalent to the summand:

The definite sum is given as the difference of indefinite sums:

Multivariate indefinite summation:

Mixes of indefinite and definite summation:

Use GenerateConditions to get the conditions under which the answer is true:

Refine or simplify the resulting answer:

Use Assumptions to provide assumptions directly to Sum:

An infinite sum may not converge:

Some infinite sums can be given a finite value using Regularization:

Applying N to an unevaluated sum effectively uses NSum:

Differences of expressions with a general function:

Polynomials can be summed in terms of polynomials:

Factorial polynomials:

Exponential sequences (geometric series):

The base-2 case plays the same role for sums as base- does for integrals:

Fibonacci and LucasL are exponential sequences with base GoldenRatio:

Exponential polynomials can be summed in terms of exponential polynomials:

Rational functions can be summed in terms of rational functions and PolyGamma:

Every difference of a rational function can be summed as a rational function:

In general, the answer will involve PolyGamma:

Every rational function can be summed:

Some rational exponential sums can be summed in terms of elementary functions:

In general, the answer involves special functions:

Every rational exponential function can be summed:

Trigonometric polynomials can be summed in terms of trigonometric functions:

Multiplied by a polynomial:

Multiplied by an exponential:

Multiplied by an exponential and a polynomial:

Hypergeometric term sequences:

The DiscreteRatio is rational for all hypergeometric term sequences:

Many functions give hypergeometric terms:

Any products are hypergeometric terms:

Differences of hypergeometric terms can be summed as hypergeometric terms:

In general additional special functions are required:

Logarithmic sums:

Some ArcTan sums can be represented in terms of ArcTan:

Similarly for ArcCot sums:

Some trigonometric sums with exponential arguments have trigonometric representations:

Products of PolyGamma and other expressions:

HarmonicNumber and Zeta behave like PolyGamma sequences:

GammaRegularized sums:

BetaRegularized sums:

Q-polynomial functions:

Multi-basic q-polynomial functions:

Mixed multi-basic q-polynomial functions:

Q-rational functions:

In general QPolyGamma is needed to represent the solution:

Rational functions of hyperbolic functions can be reduced to q-rational sums:

Q-hypergeometric terms:

Holonomic sequences generalize hypergeometric term sequences:

Any holonomic sequence can be summed:

Many special functions are holonomic:

Periodic sequences:

Periodic multiplied with a summable sequence:

Telescoping sequences:

Polynomials can be summed in terms of polynomials:

Polynomial exponentials can be summed in terms of polynomial exponentials:

Get the conditions for summability:

Rational functions can always be summed:

In general RootSum expressions are needed:

Some rational exponential functions can be summed as rational exponentials:

In general LerchPhi is required for the result:

The infinite sum is often simpler:

Trigonometric polynomials can be summed in terms of trigonometric functions:

Multiplied by a polynomial:

Multiplied by a rational function:

Multiplied by an exponential:

Logarithms of polynomials and rational functions can always be summed:

In the infinite case there is also convergence analysis:

Get the conditions for convergence:

Some hypergeometric term sums can be summed in the same class:

In general HypergeometricPFQ functions are needed:

Products of PolyGamma and other expressions:

Combining with rational and rational exponential:

Products of Zeta and HarmonicNumber with other expressions:

These are typically called Euler sums:

GammaRegularized sums:

BetaRegularized sums:

ChebyshevU sums:

ChebyshevT sums:

StirlingS1 along columns, rows and diagonals multiplied by other expressions:

Similarly for StirlingS2:

Periodic sequences multiplied by other expressions:

Infinite sums are often simpler:

Telescoping sums:

Elementary functions of several variables:

Double hypergeometric term sum:

Use Assumptions to obtain a simpler answer for an indefinite logarithmic sum:

Generate conditions required for the sum to converge:

The summand in this rational sum is singular for some values of the parameter :

Generate an arbitrary constant for an indefinite sum:

The default value for the arbitrary constant is 0:

Different methods may produce different results:

The results should be equivalent:

Many sums may not converge:

By using Regularization, many sums can be given an interpretation:

Whenever a sum converges, the regularized value is the same:

By default, convergence testing is performed:

Without convergence testing, divergent sums may return an answer:

Find expressions for the sums of powers of natural numbers:

Verify a well-known identity:

Compute the sum of a finite geometric series:

Compute the sum of an infinite geometric series:

Find the sum and radius of convergence for a power series:

Study the properties of Pascal's triangle:

The sum of the numbers of any row in Pascal's triangle is a power of 2:

The alternating sum of the numbers in any row of Pascal's triangle is 0:

The sum of the squares of the numbers in the n^th row of Pascal's triangle is Binomial[2n,n]:

The mean and variance for a Poisson distribution are both equal to the Poisson parameter:

Compute a Riemann sum approximation:

Compute an approximate value for π using Ramanujan's formula:

Find the generating function for CatalanNumber:

Construct a Taylor approximation for functions: