Chapter 4 - Functions
Page 14 of 15
|Functions - Summation Functions - Contents|
Summation functions can examine other functions using the following form: the summation of a function f(x) from x equals a to x equals b. These functions use addition to sum up a series of numbers by solving the integer variable x from numeric constant a to numeric constant b. Summations are used to examine continuity since a function is considered to be continuous at a point p, if and only if, the limit as x approaches p is the function's value at the point p. Infinite summations can be created that look at a function as the independent variable climbs towards infinity. An infinite summation series is divergent if the summation also tends towards infinity. It cannot be solved because the sum expands without boundaries. An infinite summation series is convergent if the limit of the function approaches a number or zero as x approaches infinity. This is because the summation terms will tend towards zero and the sum will converge to a solution. This can be helpful when mapping a function where the line at x goes to infinity, but still converges to a y solution value.
|Functions - Summation Functions - Examples|
y = x² + 1,
y equals the summation of the integers from one to eight of the function x squared plus one,
which is equal to (1² + 1) + (2² + 1) + (3² + 1) + (4² + 1) + (5² + 1) + (6² + 1) + (7² + 1) + (8² + 1)
and is equal to 2 + 5 + 10 + 17 + 26 + 37 + 50 + 65 = 212.
In this function as x approaches infinity this summation approaches infinity.
y = 1/(x + 1),
y equals the summation of the integers from one to eight of the function one divided by the quantity x plus one,
which is equal to (1/2) + (1/3) + (1/4) + (1/5) + (1/6) + (1/7) + (1/8) + (1/9) = 1 approximately,
as x approaches infinity this summation approaches closer to the number one as the limit converges to zero.
Limit and the summation function:
The summation above converges to one because
y = 1/(x + 1) approaches zero as x approaches infinity.
|Functions - Sections - Chapters|
|1 - Function Definition||2 - Function Notation||3 - Function Domain|
|4 - Function Range||5 - Composite Functions||6 - Inverse Functions|
|7 - Linear Functions||8 - Power Functions||9 - Quadratic Functions|
|10 - Logarithmic Functions||11 - Exponential Functions||12 - Factorial Functions|
|13 - Limit Functions||14 - Summation Functions||15 - Percentage Functions|
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