What is a geometric series?
If we add the numbers in a geometric sequence, we end up with a geometric series. A
geometric series is obtained when each term is determined from the preceeding one
by multiplying by a common ratio; there is a constant ratio between terms. For exam-
ple, 1 1/2 1/4 1/8 ..., and so on, is a geometric series because each term is
determined by multiplying the preceding term by 1/2. To find the sum of a geometric
series, the formula is: Sum a(rn1) / (r1) or a(1 rn) / (1 r), in which ais
the first term, ris the common ratio, and nis the number of terms.
For example, to find the sum of the first six terms of a series represented by 2 6
18 54 162 486, define a2; r3; and n6. Substitute the numbers:
Sum 2(3^6 1) / 3 ^1 ^729 ^1 728. We could also have determined this num-
ber based on the first few numbers, such as 2 6 18 54, as long as we knew the
common ratio, the first number, and how many numbers in the series we wanted to
add. This is something that can easily be determined based on just these four numbers.
CALCULUS BASICS
What is “the” calculus?
“The” calculus is a branch of mathematics that deals with functions; another name for
calculus is infinitesimal analysis. It evaluates constantly changing quantities, such as
velocity and acceleration; values interpreted as slopes of curves; and the area, volume,
and length objects bounded by curves (remember, “curves” can also mean straight
lines). It involves infinite processes that lead to passage to a limit,or the approaching
of an ultimate, usually desired value. The tools of the calculus include differentiation
(differential calculus, or finding a derivative) and integration (integral calculus, or find-
ing the indefinite integral), both of which are foundations for mathematical analysis.
What is a limitin calculus?
A limit is a fundamental concept in calculus. Unlike a limit mentioned above (as in a series
or sequence), a limit of a function in calculus takes on a somewhat different meaning. In
particular, a limit of a function can be described as the following: If f(x) is a function
defined around a point c(but may not be at citself), the formal limit equation becomes:
lim f (x) = L
x"cThus, the number Lis called the limit of f(x) when xgoes to c. 217