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#### Search: #### Nusum

Function: nusum (<expr>, <x>, <i_0>, <i_1>) Carries out indefinite hypergeometric summation of <expr> with respect to <x> using a decision procedure due to R.W. Gosper. <expr> and the result must be expressible as products of integer powers, factorials, binomials, and rational functions.

The terms "definite" and "indefinite summation" are used analogously to "definite" and "indefinite integration". To sum indefinitely means to give a symbolic result for the sum over intervals of variable length, not just e.g. 0 to inf. Thus, since there is no formula for the general partial sum of the binomial series, `nusum` cant do it.

`nusum` and `unsum` know a little about sums and differences of finite products. See also `unsum`.

Examples:

`          (%i1) nusum (n*n!, n, 0, n);`

```          Dependent equations eliminated:  (1)
(%o1)                     (n + 1)! - 1
(%i2) nusum (n^4*4^n/binomial(2*n,n), n, 0, n);
4        3       2              n
2 (n + 1) (63 n  + 112 n  + 18 n  - 22 n + 3) 4      2
(%o2) ------------------------------------------------ - ------
693 binomial(2 n, n)                 3 11 7
(%i3) unsum (%, n);
4  n
n  4
(%o3)                   ----------------
binomial(2 n, n)
(%i4) unsum (prod (i^2, i, 1, n), n);
n - 1
/===\
! !   2
(%o4)              ( ! !  i ) (n - 1) (n + 1)
! !
i = 1
(%i5) nusum (%, n, 1, n);```

Dependent equations eliminated: (2 3) n /===\ ! ! 2

`          (%o5)                      ! !  i  - 1`
`                                     ! !`
`                                    i = 1`

```(%o1)                                true
(%i2) ```

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##### nusum

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##### nusum

nusum(i*i*i,i,1,n);

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