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10.9 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 valid 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 algebraic simplification, evaluate to an integer.

If the number of iterations according to the above formula does not come out to an integer, the sum is invalid 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]’.