Geometric Series?

Question

I need help remember what exactly the geometric series is and how it works? can someone please explain it and give an example with real numbers. thank you!!!

Answer 1

1+2+4+8+16 .... +2n ....is a gemetric series with a common
common multiplier of 2. Each term is 2 times the previous term. The common multiplier is usualy called r.

The series 1+1/2+1/4+1/8+ ... has r =1/2
The series 1+3/2+9/4+27/8 +... has r= 3/2
The series 1-2/3+4/9-8/27+... has r = -2/3

If |r|<1, then the series will converge to a value of 1/(1-r).
If r=> 1 the series is divergent and it's sum increases without limit.

Answer 2

Geometric sequences are when you take a number, then multiply by a number each time (called the common ratio, r.) A series is a sum of a sequence.

ex.) 1+2+4+8+16+32.

This example has r=2. It also has 6 terms. Many series are infinite.

r is often less than 1 (and it can be negative.)

ex2) 6 - 2 + 2/3 - 2/9 + 2/27 - ...

This is an infinite series with a1=6 (the first term) , n=infinity (n is the number of terms, and this goes on and on,) and r= negative 1/3.

If r is between -1 and 1, then the series converges and equals a1/(1-r).

Enjoy!

Answer 3

In a geometric series, successive terms are always in the same ratio. Examples are:

1 2 4 8 16 (ratio = 2)

1 1/2 1/4 1/8 1/16 (ratio = 1/2)

Answer 4

In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed non-zero number called the common ratio. For example, the sequence 2, 6, 18, 54, ... is a geometric progression with common ratio 3 and 10, 5, 2.5, 1.25, ... is a geometric sequence with common ratio 1/2. The sum of the terms of a geometric progression is known as a geometric series.

Thus, the general form of a geometric sequence is


a, a*r, a * r^2, ........., a* r^n ............

link for knowing more:

Answer 5

In geometirc series, ratio of n+1 terms to n terms is always same.

For example,
geometric series : 3, 6, 12, 24,...
6/3 = 12/6 = 24/12 = 2!!!

If a = initial number
r = ratio of n+1 terms to n terms

n th term = ar^(n-1)

sum of n th terms,
Sn = a + ar + ar^2 + ... + ar^(n-1) ---(1)

(1) x r
rSn = ar + ar^2 + ar^3 + ...+ ar^(n) ---(2)

(2)-(1)
rSn -Sn = ar^(n) - a
Sn = a[r^(n)-1] / [r-1]