TruncateSum

TruncateSum[sexpr,n]

truncates each Sum in sexpr to have at most n terms.

TruncateSum[sexpr,{m,n,}]

truncates each multiple Sum in sexpr using the iterative specification {m,n,}.

Details and Options

  • TruncateSum is typically used to truncate symbolic solutions involving infinite sums to finite sums, making it easy to numerically evaluate such approximations.
  • The sum expression sexpr can have any combination of unevaluated and Inactive sums.
  • TruncateSum will truncate large positive or negative summation limits according to the following:
  • , if
    , if
  • The following options can be given:
  • ActivateResultTruewhether to Activate the result
    WorkingPrecision Automaticthe precision used in internal computations

Examples

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Basic Examples  (2)

Truncate an infinite sum to its first 12 terms:

Truncate an inactive sum:

Avoid activating the inactive sum:

Activate the result:

Scope  (13)

Basic Uses  (3)

Truncate an infinite sum to its first 10 terms:

Truncate an infinite sum to a sum with a symbolic upper limit:

For unevaluated sums, TruncateSum directly evaluates the truncated sum:

Finite Sums  (3)

Truncate a finite sum:

Truncate a sum with symbolic upper limit:

Truncate a sum with symmetric upper and lower limits:

Infinite Sums  (4)

Truncate an infinite sum:

Truncate a sum with doubly infinite limits:

Truncate a sum with as its lower limit:

Truncate a polynomial with infinite sum coefficients:

Multiple Sums  (2)

Truncate a doubly infinite sum:

Specify the maximum number of total terms for each sum:

Truncate only the first sum:

Truncate only the second sum:

Truncate a double sum:

Specify the maximum number of individual terms:

Inactive Sums  (1)

For inactive sums, TruncateSum tries to evaluate the truncated sum:

Use ActivateResultFalse to avoid activating the inactive sum:

Activate the result:

Options  (1)

WorkingPrecision  (1)

Truncate a double sum:

Truncate the sum using 20-digit precision arithmetic:

Applications  (10)

Differential Equations  (4)

Solve the Dirichlet problem for the wave equation on a finite interval:

The solution is an infinite trigonometric series:

Extract the first three terms from the Inactive sum:

Solve the Dirichlet problem for the wave equation in a rectangle:

The solution is a doubly infinite trigonometric series:

Extract a few terms from the Inactive sums:

Solve the Dirichlet problem for the heat equation on a finite interval:

The solution is a Fourier sine series:

Truncate the Inactive sum:

Solve the initial value problem for a Schrödinger equation with Dirichlet boundary conditions:

Define a family of partial sums of the solution:

For each k, uk satisfies the differential equation:

The boundary conditions are also satisfied:

The initial condition is only satisfied for u, but there is rapid convergence at t==2:

Difference Equations  (1)

Solve a difference equation:

Extract the solution for :

Asymptotics  (3)

Compute the power series expansion of around 0:

Obtain the first seven nonzero terms in the series:

Compute the power series expansion of around 1:

Obtain the first five terms in the series:

Truncate the series for the Hypergeometric1F1 function:

Compare the results with the built-in Hypergeometric1F1 function:

Inverse Laplace Transform  (2)

Calculate the inverse Laplace transform of a function:

Truncate the sum and plot the result:

The inverse Laplace transform of this function is a piecewise function:

Truncate and plot the result:

Properties & Relations  (4)

TruncateSum[expr,n] truncates each sum in the expression to have at most n terms:

TruncateSum truncates both limits of doubly infinite sums:

TruncateSum truncates only the lower limit of a sum from to a finite value:

TruncateSum activates all inactive sums in the expression:

ActivateResult False can be used to avoid activation:

Possible Issues  (1)

The number of terms in the truncated sum is assumed to be an integer:

Wolfram Research (2023), TruncateSum, Wolfram Language function, https://reference.wolfram.com/language/ref/TruncateSum.html.

Text

Wolfram Research (2023), TruncateSum, Wolfram Language function, https://reference.wolfram.com/language/ref/TruncateSum.html.

CMS

Wolfram Language. 2023. "TruncateSum." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/TruncateSum.html.

APA

Wolfram Language. (2023). TruncateSum. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/TruncateSum.html

BibTeX

@misc{reference.wolfram_2024_truncatesum, author="Wolfram Research", title="{TruncateSum}", year="2023", howpublished="\url{https://reference.wolfram.com/language/ref/TruncateSum.html}", note=[Accessed: 25-June-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_truncatesum, organization={Wolfram Research}, title={TruncateSum}, year={2023}, url={https://reference.wolfram.com/language/ref/TruncateSum.html}, note=[Accessed: 25-June-2024 ]}