A definite sum over a finite range:

Using step size 2:

Using a finite list of elements:

Plot the sequence and its partial sums:

A multiple sum over finite ranges:

Using a different step size:

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:

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:

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:

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:

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:

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

Similarly for

StirlingS2:

Periodic sequences multiplied by other expressions:

Infinite sums are often simpler:

Elementary functions of several variables: