Summations
==========

The `a +' (`calc-summation') [`sum'] command computes the sum of a
formula over a certain range of index values.  The formula is taken
from the top of the stack; the command prompts for the name of the
summation index variable, the lower limit of the sum (any formula),
and the upper limit of the sum.  If you enter a blank line at any of
these prompts, that prompt and any later ones are answered by reading
additional elements from the stack.  Thus, `' k^2 RET ' k RET 1 RET 5
RET a + RET' produces the result 55.

The choice of index variable is arbitrary, but it's best not to
use a variable with a stored value.  In particular, while
`i' is often a favorite index variable, it should be avoided
in Calc because `i' has the imaginary constant `(0, 1)'
as a value.  If you pressed `=' on a sum over `i', it would
be changed to a nonsensical sum over the "variable" `(0, 1)'!
If you really want to use `i' as an index variable, use
`s u i RET' first to "unstore" this variable.
(See Storing Variables.)

A numeric prefix argument steps the index by that amount rather than
by one.  Thus `' a_k RET C-u -2 a + k RET 10 RET 0 RET' yields `a_10 +
a_8 + a_6 + a_4 + a_2 + a_0'.  A prefix argument of plain `C-u' causes
`a +' to prompt for the step value, in which case you can enter any
formula or enter a blank line to take the step value from the stack.
With the `C-u' prefix, `a +' can take up to five arguments from the
stack: The formula, the variable, the lower limit, the upper limit,
and (at the top of the stack), the step value.

Calc knows how to do certain sums in closed form.  For example,
`sum(6 k^2, k, 1, n) = 2 n^3 + 3 n^2 + n'.  In particular,
this is possible if the formula being summed is polynomial or
exponential in the index variable.  Sums of logarithms are
transformed into logarithms of products.  Sums of trigonometric
and hyperbolic functions are transformed to sums of exponentials
and then done in closed form.  Also, of course, sums in which the
lower and upper limits are both numbers can always be evaluated
just by grinding them out, although Calc will use closed forms
whenever it can for the sake of efficiency.

The notation for sums in algebraic formulas is `sum(EXPR, VAR, LOW,
HIGH, STEP)'.  If STEP is omitted, it defaults to one.  If HIGH is
omitted, LOW is actually the upper limit and the lower limit is one.
If LOW is also omitted, the limits are `-inf' and `inf', respectively.

Infinite sums can sometimes be evaluated: `sum(.5^k, k, 1, inf)'
returns `1'.  This is done by evaluating the sum in closed form (to
`1. - 0.5^n' in this case), then evaluating this formula with `n' set
to `inf'.  Calc's usual rules for "infinite" arithmetic can find the
answer from there.  If infinite arithmetic yields a `nan', or if the
sum cannot be solved in closed form, Calc leaves the `sum' function in
symbolic form.  See Infinities.

As a special feature, if the limits are infinite (or omitted, as
described above) but the formula includes vectors subscripted by
expressions that involve the iteration variable, Calc narrows the
limits to include only the range of integers which result in legal
subscripts for the vector.  For example, the sum `sum(k
[a,b,c,d,e,f,g]_(2k),k)' evaluates to `b + 2 d + 3 f'.

The limits of a sum do not need to be integers.  For example,
`sum(a_k, k, 0, 2 n, n)' produces `a_0 + a_n + a_(2 n)'.  Calc
computes the number of iterations using the formula `1 + (HIGH - LOW)
/ STEP', which must, after simplification as if by `a s', evaluate to
an integer.

If the number of iterations according to the above formula does not
come out to an integer, the sum is illegal and will be left in
symbolic form.  However, closed forms are still supplied, and you are
on your honor not to misuse the resulting formulas by substituting
mismatched bounds into them.  For example, `sum(k, k, 1, 10, 2)' is
invalid, but Calc will go ahead and evaluate the closed form solution
for the limits 1 and 10 to get the rather dubious answer, 29.25.

If the lower limit is greater than the upper limit (assuming a
positive step size), the result is generally zero.  However, Calc only
guarantees a zero result when the upper limit is exactly one step less
than the lower limit, i.e., if the number of iterations is -1.  Thus
`sum(f(k), k, n, n-1)' is zero but the sum from `n' to `n-2' may
report a nonzero value if Calc used a closed form solution.

Calc's logical predicates like `a < b' return 1 for "true" and 0 for
"false."  See Logical Operations.  This can be used to advantage
for building conditional sums.  For example, `sum(prime(k)*k^2, k, 1,
20)' is the sum of the squares of all prime numbers from 1 to 20; the
`prime' predicate returns 1 if its argument is prime and 0 otherwise.
You can read this expression as "the sum of `k^2', where `k' is
prime."  Indeed, `sum(prime(k)*k^2, k)' would represent the sum of
*all* primes squared, since the limits default to plus and minus
infinity, but there are no such sums that Calc's built-in rules can do
in closed form.

As another example, `sum((k != k_0) * f(k), k, 1, n)' is the sum of
`f(k)' for all `k' from 1 to `n', excluding one value `k_0'.  Slightly
more tricky is the summand `(k != k_0) / (k - k_0)', which is an
attempt to describe the sum of all `1/(k-k_0)' except at `k = k_0',
where this would be a division by zero.  But at `k = k_0', this
formula works out to the indeterminate form `0 / 0', which Calc will
not assume is zero.  Better would be to use `(k != k_0) ? 1/(k-k_0) :
0'; the `? :' operator does an "if-then-else" test: This expression
says, "if `k != k_0', then `1/(k-k_0)', else zero."  Now the formula
`1/(k-k_0)' will not even be evaluated by Calc when `k = k_0'.

The `a -' (`calc-alt-summation') [`asum'] command computes an
alternating sum.  Successive terms of the sequence are given
alternating signs, with the first term (corresponding to the lower
index value) being positive.  Alternating sums are converted to normal
sums with an extra term of the form `(-1)^(k-LOW)'.  This formula is
adjusted appropriately if the step value is other than one.  For
example, the Taylor series for the sine function is `asum(x^k / k!, k,
1, inf, 2)'.  (Calc cannot evaluate this infinite series, but it can
approximate it if you replace `inf' with any particular odd number.)
Calc converts this series to a regular sum with a step of one, namely
`sum((-1)^k x^(2k+1) / (2k+1)!, k, 0, inf)'.

The `a *' (`calc-product') [`prod'] command is the analogous way to
take a product of many terms.  Calc also knows some closed forms for
products, such as `prod(k, k, 1, n) = n!'.  Conditional products can
be written `prod(k^prime(k), k, 1, n)' or `prod(prime(k) ? k : 1, k,
1, n)'.

The `a T' (`calc-tabulate') [`table'] command evaluates a formula at a
series of iterated index values, just like `sum' and `prod', but its
result is simply a vector of the results.  For example, `table(a_i, i,
1, 7, 2)' produces `[a_1, a_3, a_5, a_7]'.