How to | Evaluate Infinite Sums and Products

In calculus, infinite sums and products can pose a challenge to manipulate by hand. Mathematica can evaluate a huge number of different types of sums and products with ease.

Use Sum to set up the classic sum , with the function to sum over as the first argument. Use Mathematica's usual range notation as the second argument:

In[247]:=
Click for copyable input
Out[247]=

This also works for finite sums like :

In[249]:=
Click for copyable input
Out[249]=

Use 1. to get the decimal representation:

In[250]:=
Click for copyable input
Out[250]=

This checks that :

In[251]:=
Click for copyable input
Out[251]=

Some functions have an infinite sum representation, and Mathematica will recognize these. For example :

In[252]:=
Click for copyable input
Out[252]=
    

Many functions have product representations as well, and Mathematica will even recognize these.

Use Product to check , a function found by the mathematician Euler. The arguments of Product have the same form as Sum:

In[253]:=
Click for copyable input
Out[253]=

Even more abstract functions will be recognized; the Product representation of involves the set of prime numbers:

In[254]:=
Click for copyable input
Out[254]=
New to Mathematica? Find your learning path »
Have a question? Ask support »